Modifying spiking precision in conductance-based …leaky integrate-and-fire (LIF) model and an...

2013

Network: Computation in Neural SystemsMarch 2013; 24(1): 1–26

Modifying spiking precision in conductance-basedneuronal models

CYRUS P. BILLIMORIA1, RALPH A. DICAPRIO2,

ASTRID A. PRINZ3, VICTOR QUINTANAR-ZILINSKAS4#, &

JOHN T. BIRMINGHAM4

1Hearing Research Center, Department of Biomedical Engineering, Boston University,

Boston, MA, 2Department of Biological Sciences, Ohio University, Athens, OH,3Department of Biology, Emory University, Atlanta, GA, and 4Department of Physics,

Santa Clara University, Santa Clara, CA

(Received 13 November 2012; revised 13 December 2012; accepted 14 December 2012)

AbstractThe temporal precision of a neuron’s spiking can be characterized by calculating its ‘‘jitter,’’defined as the standard deviation of the timing of individual spikes in response to repeatedpresentations of a stimulus. Sub-millisecond jitters have been measured for neurons in avariety of experimental systems and appear to be functionally important in some instances.We have investigated how modifying a neuron’s maximal conductances affects jitter using theleaky integrate-and-fire (LIF) model and an eight-conductance Hodgkin-Huxley type (HH8)model. We observed that jitter can be largely understood in the LIF model in terms of theneuron’s filtering properties. In the HH8 model we found the role of individual conductancesin determining jitter to be complicated and dependent on the model’s spiking properties.Distinct behaviors were observed for populations with slow (<11.5 Hz) and fast (>11.5 Hz)spike rates and appear to be related to differences in a particular channel’s activity at timesjust before spiking occurs.

Keywords: coding, jitter, temporal precision, conductance-based model

Introduction

How well a sensory neuron can translate a stimulus into a neural signal depends

both on how often and how accurately the neuron spikes. More frequent spiking, i.e.

Correspondence: John T. Birmingham, Department of Physics, Santa Clara University, 500 El Camino Real, Santa

Clara, CA 95053. Tel: (408) 551-7185. Fax: (408) 554-6965. E-mail: [email protected]#Present address: Department of Biomedical Engineering, University of California, Irvine, Irvine, CA

ISSN 0954-898X print/ISSN 1361-6536 online/03/010001–26 � 2013 Informa UK Ltd.

DOI: 10.3109/0954898X.2012.760057

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a larger average spike rate, allows the stimulus to be sampled more often and finer

temporal features to be described, regardless of whether the coding is better

described as a rate or a temporal code. In the case of a temporal code, more precise

spiking establishes the times of those features more definitively. This precision can

be quantified by making repeated presentations of a stimulus, recording the

corresponding spike trains and calculating the ‘‘jitter,’’ defined as the standard

deviation of the timing of individual spikes.

Sub-millisecond jitters have been measured for neurons in a variety of systems

ranging from primary sensory receptors in invertebrates (Billimoria et al. 2006;

Rokem et al. 2006; DiCaprio et al. 2007) to the vertebrate CNS (Mainen and

Sejnowski 1995; Beierholm et al. 2001). The necessity of precise timing is readily

apparent in systems that use the relative times of spikes to compute locational

information (Carr and Konishi 1990; Heiligenberg 1991; Kuwabara and Suga

1993). Neurons in auditory pathways, for example, show phase-locking at

frequencies up to 3–5 kHz for some mammals (Goldberg and Brownell 1973;

Crowe et al. 1978; Sullivan and Konishi 1984) and up to 9 kHz for barn owls (Carr

and Konishi 1990), and the ability to perceive interaural time differences of �10 ms

can be inferred from behavioral experiments. However, the functional significance

of the jitter associated with individual neurons, particularly when the jitter is much

smaller than the time scales associated with stimuli or postsynaptic integration,

remains an area of debate. Mackevicius et al. (2012) recently argued that

millisecond and sub-millisecond jitters in primate mechanoreceptive afferents play

an important role in the perception of high-frequency (>100 Hz) skin vibrations.

This conclusion was reached by comparing the predictions of a temporal code,

assuming different spike-timing precisions, to actual behavioral results. Information

theory (de Ruyter van Steveninck et al. 1997; Rolls and Treves 2011) provides a

purely computational approach to this issue and has been used to support the notion

that sub-millisecond jitter can affect a neuron’s ability to encode information

(Reinagel and Reid 2000; Rokem et al. 2006), even for a stimulus that varies on a

time scale that is orders of magnitude larger than the jitter (Butts et al. 2007;

Nemenman et al. 2008).

It is well known that a neuron’s spike rate depends both on stimulus features (e.g.

amplitude, frequency, waveform) and on membrane properties of the neuron, in

particular the complement and concentrations of ion channels that govern cellular

excitability. In contrast, the relative importance of the factors that influence the jitter

and how they interact are not well established. From a coding perspective,

correlations between rate and jitter could be very important. For example, if jitter

and rate were positively correlated, a process that increased the spiking could permit

more temporal features to be encoded in the spike train at the expense of knowledge

of their exact times. Whether this, overall, would lead to improved coding would

require a detailed calculation. All else being equal, more frequent spiking and

increased jitter would have opposite effects on the neuron’s information transfer rate

(in bits/sec). The trade-off between bits/spike and bits/sec, and the relationship

between coding, rate, temporal precision and energy efficiency has recently been

analyzed quantitatively (Berger and Levy 2010).

To the best of our knowledge, there have been only two experimental studies that

have focused on the rate-jitter relationship (Tang et al. 1997; Billimoria et al. 2006).

In the more recent of these, the authors described experiments in which two

2 C. P. Billimoria et al.

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different crustacean stretch receptors were mechanically stimulated using filtered

white noise waveforms. The inquiry produced two notable results. First, spike rate

and jitter in individual stretch receptor afferents of either type could be modified by

application of neuromodulatory substances. In one of the receptors, intracellular

recordings were possible, and it was observed that the effects on rate and jitter were

accompanied by changes in the cell’s membrane resistance. Second, the authors

documented significant correlations between spike rate and jitter for each stretch

receptor, with the sign of the correlation coefficient being positive in one case and

negative in the other. The sum of these experimental results suggested that jitter,

like spike rate, could be modified through changes in the maximal conductances of

one or more ion channel types in a neuron.

In this paper we describe an investigation of how channel type and number can

affect a neuron’s jitter. In an experimental system there is little or no control over

noise sources or a neuron’s conductances, and so we have taken a computational

approach, utilizing two very different models to explore this issue. Initially we

employed a leaky integrate-and-fire model to demonstrate how simply changing the

membrane conductance can affect jitter for various stimulus/noise combinations.

Then utilizing a database of approximately 10 000 parameterizations of an eight-

channel conductance-based model, we explored how jitter and rate interact in a

more realistic biophysical model neuron. Here we found a much richer relationship

between jitter and rate than in the simpler model. We discovered that certain

channel types can be particularly important for determining jitter and that the

magnitude and even the sign of the effect can depend on the spiking properties of

the neuron.

Methods

Leaky integrate-and-fire (LIF) model

The leaky integrate-and-fire (LIF) model describes a generic neuron with passive

membrane properties that spikes when its membrane potential V reaches a threshold

Vthresh. For V < Vthresh the membrane potential evolves according to

CdV

dt¼

EL � Vð Þ

Rþ Ie ð1Þ

where C and R are the membrane capacitance and resistance, respectively, EL is the

reversal potential, and Ie is a time-varying injected current. Our canonical LIF

neuron had values of R¼ 10 M� (G¼R�1¼ 0.1 mS) and C¼ 1 nF, corresponding to

a membrane time constant of �¼RC¼ 10 ms. After the neuron spikes, V

immediately resets to Vreset below Vthresh. In our model Vthresh¼�50 mV,

Vreset¼�70 mV, and EL¼�65 mV. In real neurons, the probability of spiking is

significantly reduced immediately after a spike is produced, but the basic LIF model

has no such built-in refractory period. To limit the rates and to more closely mimic

the experimental behavior being modeled, we imposed a 3 ms hard refractory period

after each spike, during which V was held at Vreset. At the end of the refractory

period, Equation 1 again described the evolution of V.

Modifying spiking precision in model neurons 3

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The LIF model is computationally inexpensive (Izhikevich 2004), and in spite of

its simplicity, a version of the model with a moving threshold (Kobayashi et al.

2009) recently won a competition (INCF, 2009) in which the ability to predict the

timing of spikes was tested (Gerstner and Naud 2009). To numerically integrate the

model, we used Matlab (Ver. 6.1; The Mathworks; Natick, MA) to implement the

exponential Euler method using a time step of 50ms. We found that reducing the

time step to 25 ms did not change our results.

Eight-conductance Hodgkin-Huxley type (HH8) model

To simulate more biophysically realistic neurons, we used a database of parame-

terizations of a conductance-based model that was developed by some of us and that

has been previously described (Prinz et al. 2003a). The model consisted of a single

compartment that contains eight Hodgkin-Huxley type membrane conductances

and an intracellular calcium buffer. The eight conductances were fast sodium (Na),

delayed rectifier potassium (Kd), fast transient potassium (A), calcium-dependent

potassium (KCa), fast and slow calcium (CaT, CaS), hyperpolarization-activated

inward (H), and voltage-independent leak (L). The mathematical description of

each conductance was based upon experimental data from the California spiny

lobster, Panulirus interruptus (Turrigiano et al. 1995). The database we used was

constructed by varying the maximal value of each conductance over six levels (0, 1,

2, 3, 4 and 5 times a canonical value). This was done independently for each of the

eight conductances to produce a total of 68 or 1 679 616 versions of the model.

In this model the membrane potential evolves according to

CdV

dt¼ �

XjIj þ Ie ð2Þ

where each membrane current is described by Ij ¼ �gjmpj

j hj ðV � Ej Þa. Here �gj is the

maximal conductance, Ej is the reversal potential with j¼ {Na, Kd, A, KCa, CaT,

CaS, H and L}, and a is the membrane surface area (0.628� 10�3 cm2). The

equation describing intracellular calcium concentration [Ca2þ] and the expressions

for the voltage dependence of the activation parameters mj and hj and the exponents

pj are given in Prinz et al. (2003a; 2003b). All differential equations were integrated

in the C programming language using Euler’s method with a 25 ms time step.

Prinz et al. (2003a) previously showed that neurons in this database exhibit a

variety of spontaneous spiking behaviors including bursting, tonic firing, irregular

firing and silent. Most of the approximately 264 thousand spontaneous tonic spikers

(constant interspike interval) generated narrow spikes, while a smaller population

produced single spikes that were followed by a broad voltage shoulder and

corresponded to one-spike bursters (Prinz et al. 2003a). Hong et al. (2008) further

showed that the tonic, narrow spikers consisted of two separate subpopulations: one

with a spike rate >11.5 Hz (‘‘fast spikers’’) and one with a rate <11.5 Hz

(‘‘slow, narrow spikers’’). For our investigation, we chose to use parameterizations

of the model that fired tonically both spontaneously and in response to current

injection. We found 263,114 instances in which the model did this in response to 0,

3 and 6 nA constant current injections; nearly all of the tonic, narrow spikers

satisfied this requirement. Of these we picked approximately ten thousand

parameterizations (9946 to be precise) at random for our test set. This number


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achieved a balance between maximal sampling of the set of tonic spikers and total

run time, given the computing power available.

Although the primary transduction process in a mechanoreceptor has

been modeled mathematically in other crustacean systems (e.g. Swerup

and Rydqvist 1996), we chose not to include mechanotransduction or

stretch-sensitive conductances in our model both for simplicity and in the hope

that our results would be more easily interpreted and generalizable to non-sensory

neurons.

Current injection of deterministic and random waveforms

In the experimental investigation that motivated this work, the coxobasal

chordotonal organ (CBCTO) in the crab Carcinus maenas was stimulated

mechanically using a waveform constructed from Gaussian white noise (GWN)

that was low-pass filtered at 140 Hz (Billimoria et al. 2006). Preliminary

(unpublished) data indicated that the transform from mechanical stimulus to

voltage waveform is nonlinear, but that the frequency range is unaltered. We

mimicked this stimulus in our computational investigations by injecting each of our

model versions with a simulated current waveform (the ‘‘signal’’) possessing this

frequency spectrum.

In experiments in which CBCTO was stimulated mechanically, DiCaprio et al.

(2007) measured a mean jitter of 0.539 ms, with a range from 0.4 to 0.65 ms. When

spikes were evoked by current injection, the mean jitter was measured to be

0.098 ms, and the authors concluded that, although some jitter is probably

associated with the ion channels mediating the production of action potentials, the

majority of it likely originates in the biomechanical properties of the receptor,

in transduction channels, and/or in coupling to the receptor structure

(DiCaprio et al. 2007).

In our simulations we approximated this noise as a random current waveform,

with characteristics described below, added to the deterministic signal to induce

jitter. Simple additive noise is only a first-order approximation of the real situation

but has been frequently used in mathematical studies of reliability and spike timing.

The power spectral density (PSD) of the current noise associated with a two-state

(open/closed) channel (for example, a transduction or afferent channel) can be

shown to be proportional to 1/(1þ (f/fc)2), where the cutoff frequency fc is inversely

proportional to the activation time constant of the channel (DeFelice 1981; Diba

et al. 2004). For f « fc the above expression is approximately constant, and for f » fc it

rolls off as f�2. The aggregate noise due to multiple channel types contributes

additively to the PSD, and, in practice, the current PSD can roll off with a different

frequency dependence due to cable properties and the geometry of the cell (Diba

et al. 2004; Jacobson et al. 2005) or additional non-Lorentzian noise sources

(DeFelice et al. 1975). The intrinsic membrane noise in CBCTO afferents has not

been characterized, but preliminary measurements of the membrane potential noise

in these cells (R. DiCaprio, data not shown) indicated that the PSD of the noise is

very similar to that measured in hippocampal neurons by Diba et al. (2004). We

considered a few extreme cases in our simulations: In the LIF study we used for the

random current waveform a cutoff frequency of either 20 Hz or 1 kHz, frequencies

that were respectively well below or above 140 Hz. For simplicity, we cut the


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spectrum of the random waveform off sharply at the cutoff frequency using a fourth-

order Butterworth filter, rather than guessing at the exponent associated with the

roll off. In the HH8 study the random current waveform was white up to 2 kHz,

limited only by the sampling rate.

Using a random current waveform with a constant amplitude for all of the

parameterizations tested is appropriate if the noise represented by the waveform

originates upstream from the HH8 neuron itself, for example if jitter were due to

mechanotransduction or to synaptic noise. (To the best of our knowledge, CBCTO

does not receive peripheral synaptic input.) Although most of the CBCTO jitter was

likely generated during mechanotransduction, an appreciable contribution appears

attributable to ‘‘channel noise’’ associated with conductances in or near the spike

initiation zone. How channel noise can influence temporal precision in computa-

tional models of neurons has been explored in a number of systems (Rowat and

Elson 2004; Ozer et al. 2009), but we do not consider this possibility here. If

channel noise in an HH8 conductance were the source of the jitter, using a fixed

amplitude for the random current waveform would not be appropriate; the

amplitude should scale approximately as the square root of its maximal conductance

if the changes in maximal conductance reflect variations in the number of channels.

Moreover, the spectrum for the random waveform would depend on the species of

the putatively noisy channel; values of the cutoff frequency fc below 10 Hz (Cull-

Candy and Usowicz 1989; Diba et al. 2004) and above 1 kHz (White et al. 1998)

have been reported.

Data analysis, statistics and figure production

To quantify the precision of spiking in the two models, we calculated the ‘‘jitter’’

and ‘‘reliability’’ of individual ‘‘events’’ produced in response to repeated

presentations of a stimulus. Our definitions of these terms are nearly identical to

the original ones put forth by Mainen and Sejnowski (1995) and are reviewed below.

Our general method for obtaining jitter was described in detail in Billimoria et al.

(2006). In short, in each ‘‘experiment’’ a model neuron was injected with 50 or

more repeated presentations (‘‘trials’’) of an identical current signal constructed

from low-pass filtered GWN. This sort of stimulus, in which an aperiodic waveform

is repeatedly presented, has been termed ‘‘frozen noise’’ (Haas and White 2002).

For each trial, a different small random current waveform produced from filtered

GWN was added to the signal, as described in the previous section. Spike times

were defined as upward-sloping zero crossings in the resulting voltage waveform,

and a peristimulus time histogram (PSTH) (bin size of 0.5 ms) was constructed

from the times of all spikes in all trials. Events were identified in each PSTH as

features in the stimulus that reliably (at least 50% of the time) evoked a spike in each

trial, and the percentage of trials in which an event occurred was defined as its

reliability. The beginning and end of an event were defined by bins with zero spikes

in the PSTHs, and the jitter of the event was defined as the standard deviation of the

actual spike times associated with the event. For the input waveforms and high

signal-to-noise (S/N) ratios used, we found that the identification of events and

determination of jitters were insensitive to the particulars of the calculation. For

example, using a bin size of 1 ms in the LIF calculations resulted in nearly identical

results. It is worth noting that, in general, the definition of what constitutes an event


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has not been a trivial matter, particularly in experiments where there is high

background spiking and/or the events overlap each other, but that recent progress

has been made on this issue (Toups et al. 2011). In our study, after discarding the

first second of each response to eliminate transient behavior, we averaged jitter

across all events to obtain the average jitter for the experiment and then averaged

across experiments in order to place error bars on jitter measurements. To

investigate the filtering properties of the LIF model neuron, we made calculations of

the PSD of voltage waveforms using Welch’s averaged, modified periodogram

method (Matlab). Finally, to analyze the relationships between rate and jitter,

between rate and maximal conductance, and between jitter and maximal conduc-

tance, we calculated the Pearson product-moment correlation coefficient, which

varies between þ1 and �1 and is a measure of the strength of the linear dependence

between two variables. These computations were also done in Matlab. Statistical

significance is indicated on figures using the following symbols: �P < 0.05,��P < 0.01, ��P < 0.001. All uncertainties reported correspond to standard errors.

Figures were produced using SigmaPlot (v. 10; Systat Software; San Jose, CA).

Results

In the study of the CBCTO receptor in C. maenas, Billimoria et al. (2006) found

that correlated changes in membrane resistance and in the magnitude of jitter could

be altered by the application of neuromodulatory substances. This suggested that

the modification of jitter might be due in part to changes in the neuron’s filtering

properties resulting from a modified membrane time constant. We decided to

explore computationally how changing a neuron’s conductances affects jitter that

results from the injection of a random current waveform in two different single-

compartment model neurons: the leaky integrate-and-fire (LIF) neuron and an

eight-conductance Hodgkin-Huxley type (HH8) neuron.

Modification of jitter in LIF model through changes in leak conductance

The jitter of a particular event is related to the slope of the voltage waveform

(averaged over trials) just before spiking; a steeper slope narrows the window during

which noise can drive spiking forward or backward in time (Stein 1967; Bryant and

Segundo 1976; Goldberg et al. 1984; Gerstner et al. 1996; Hunter et al. 1998;

Pillow et al. 2005). For spike trains that consist of multiple events, one would expect

that average jitter should depend on the amplitudes and spectra of the signal and

noise, as well as on the signal-processing properties and excitability of the neuron.

In studies using an Aplysia motor neuron and the LIF model, Hunter and Milton

(2003) considered the spiking in response to a stimulus consisting of DCþ frozen

noise and found that the ‘‘reliability statistic’’ of the spike trains, which incorporates

both the reliability and the jitter, was dependent on the amplitude of the time-

varying portion of the stimulus. For stimuli with the smallest amplitudes, the spiking

was unreliable regardless of the frequency content of the stimulus; for intermediate

amplitudes, a high reliability statistic required the presence of a resonant frequency

in the stimulus (Hunter et al. 1998); at high amplitudes the neuron spiked


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reliably regardless of the absence or presence of the resonant frequency.

The three regimes of behavior were confirmed in simulations using an LIF model

subject to GWN.

Changing the membrane resistance of an LIF neuron would be expected to have

effects on jitter, both through modifications of filtering properties and, if in the right

regime, by changing the resonant frequency. In our study, we explored jitter in the

LIF model using a stimulus that was qualitatively most similar to the large

amplitude stimuli used in the Aplysia study. Our six-second duration deterministic

signal (Irms¼ 5 nA, zero mean) was constructed from GWN low-pass filtered at

140 Hz, to which we added a random current waveform with a rms amplitude of

0.5 nA, producing a stimulus with a S/N of 10:1. Initially, we chose a very low

frequency (20 Hz) band for the random waveform. Portions of two such stimulus

waveforms (same signalþdifferent random waveform) of 650 ms duration are

shown in the top panel of Figure 1(a), although with this S/N it is difficult to

distinguish between the stimuli. Directly below is the voltage trace elicited by one of

the stimulus waveforms, and under the voltage trace is a raster plot showing the

spike times measured in response to ten repetitions of the stimulus. The first of

these raster responses corresponds to the voltage trace, and the first two correspond

to the two stimuli shown in the top panel. Below the rasters is a PSTH (bin size

0.5 ms) showing the results of 1000 trials. Over the last five seconds of the response,

the average spike rate was 30.4 Hz, and the mean jitter averaged over all events was

0.58 ms. This jitter was very similar to the average measured in CBCTO

experiments (DiCaprio et al. 2007).

Figure 1(b) shows results when the protocol was repeated with an identical signal

and with identical noise waveforms, but with the neuron’s conductance increased

fourfold to G¼ 0.4 mS. The voltage trace in response to the first stimulus, a raster of

the first ten trials, and the PSTH for 1000 trials are shown. In comparing Figure

1(b) to Figure 1(a), three things become apparent. First, the spike rate decreased

with increased conductance, as would be expected, falling to an average rate of

18.0 Hz. Second, jitter decreased with increased conductance as evidenced by

narrower peaks in the PSTHs in Figure 1(b) and in Figure 1(d), dropping to an

average value of 0.30 ms. Third, the fluctuations in the voltage traces between spikes

appear to be of smaller amplitude and dominated by higher frequencies in Figure

1(b) than in Figure 1(a).

Increasing conductance has two effects: it causes the neuron to respond faster,

but it also results in larger amplitude stimuli being needed for the neuron to reach

threshold. We suspected that the first phenomenon was responsible for the

reduction in jitter. To address the role of the second one, we repeated the

measurements shown in Figure 1(a) but added a constant (�1 nA) hyperpolarizing

current to the time-varying stimulus used previously. The magnitude of this current

was chosen so that the average spike rate (18.6 Hz) was very similar to that for the

experiment shown in Figure 1(b). Results are shown in Figure 1(c) and 1(d). The

peaks in the PSTH (Figure 1c) were broader than when the conductance was

increased (Figure 1b), and the mean jitter measured (0.54 ms) was very similar to

that for the original experiment (Figure 1a) in which the conductance took on the

canonical value. When we repeated the experiment shown in Figure 1(c), changing

the constant current to �2 nA and then �3 nA, we obtained mean spike rates of


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Figure 1. Changes in jitter in the leaky integrate-and-fire (LIF) model neuron can beexplained by its filtering properties. (a) Response of a LIF neuron (R¼ 10 M� [G¼ 0.1 mS],C¼ 1 nF, Vthresh¼�50 mV, Vreset¼�70 mV, EL¼�65 mV, 3 ms refractory period) to currentinjection stimuli with S/N¼ 10. The injected current was a deterministic six-second duration‘‘signal’’ (5 nA rms) consisting of Gaussian white noise (GWN) low-pass filtered at 140 Hz towhich a random waveform (0.5 nA rms white noise low-pass filtered at 20 Hz) was added. Aportion of each of two particular presentations of the stimulus is shown at top. Because of thehigh S/N, the two traces are nearly indistinguishable. Below this is the voltage tracecorresponding to one of the stimuli. Below the voltage trace are rasters showing the spiketimes for ten presentations. The first of these raster responses corresponds to the voltagetrace, and the first two correspond to the two stimuli shown in the top panel. Below therasters is a PSTH (bin size 0.5 ms) summarizing the responses to 1000 presentations(‘‘trials’’) of the stimulus. Average rate¼ 30.4 Hz. Average jitter (over all events)¼ 0.58 ms.Note that the actual spike times (and not the binned times) were used to calculate jitter.

(continued)


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10.2 Hz and 4.6 Hz and mean jitters of 0.51 ms and 0.55 ms, respectively (data not

shown).

To confirm that the filtering properties of the neuron were modified by the

conductance change, we calculated the PSD of the voltage traces produced by the

LIF model. We used a single presentation of a stimulus with duration 101 seconds

and having the same characteristics as that used in Figure 1(a) and (b). Again, the

first second of the response was not used in the analysis. The PSD functions plotted

in Figure 1(e) show that the total power in the voltage signal was smaller for

G¼ 0.4mS than for G¼ 0.1 mS, consistent with its having a smaller amplitude.

However, for G¼ 0.4 mS, proportionally more of the power was found at higher

frequencies, consistent with the faster waveform seen in Figure 1(b) and with the

cutoff frequency f0¼ 1/(2�RC) increasing from approximately 16 to 64 Hz. For both

conductances, there was increased rolloff above 100 Hz due to the input current

signal itself having been low-pass filtered at 140 Hz.

We extended these measurements to investigate how jitter depends on the

spectrum of the random waveform by considering two different bandwidths and

employing the same model neuron and signal as before. For each of eight values of

the membrane conductance (multiples of the canonical value), we repeated the

protocol used to generate Figure 1(a) and (b) for random waveforms with

bandwidths of 20 and 1000 Hz, again with a 10:1 S/N. Here we used 50 rather than

1000 repetitions of a six-second stimulus to calculate the average rate and jitter for

an experiment. We then averaged the results of 25 experiments, each using a

different signal, to find the mean and standard error for the average rate and jitter for

each conductance value. Figure 1(f) and (g) plot spike rate and jitter, respectively, as

a function of the membrane conductance for the two cases considered. For both

bandwidths the spike rate decreased with increased conductance, as expected, and

the two plots, in fact, were essentially indistinguishable (Figure 1f). Overall, the

jitter was larger for the 20 Hz than for the 1000 Hz bandwidth (Figure 1g), again as

expected, because in the latter case, more of the noise power was carried by high

(continued)(b) Top: voltage trace of LIF model when G¼ 0.4 mS in response to same current injectionshown at top of (a). Middle: rasters corresponding to same ten presentations of stimulus.Bottom: PSTH summarizing responses to 1000 presentations. Average rate¼ 18.0 Hz.Average jitter¼ 0.30 ms (c) Top: voltage trace of LIF model when G¼ 0.1 mS in response tostimulus consisting of same time-varying current injection from above to which a constanthyperpolarizing current of �1 nA was added. Corresponding ten responses and PSTH below.Average rate¼ 18.6 Hz. Average jitter¼ 0.54 ms. (d) Expanded PSTH for peaks indicated by# in (a), þ in (b) and & in (c). Corresponding events show less jitter in case (þ) whereG¼ 0.4 mS than when G¼ 0.1 mS (# or &). (e) Power spectral density (PSD) calculated inresponse to current injection stimulus with same characteristics as in (a) and (b) for LIFmodel with G¼ 0.1 mS and G¼ 0.4mS. Here the stimulus was a single presentation with aduration of 101 seconds, and the last 100 seconds were used in the analysis. Dashed verticallines indicate 16 Hz (cutoff frequency of the neuron when G¼ 0.1 mS), 64 Hz (cutofffrequency when G¼ 0.4 mS), and 140 Hz (frequency at which input current signal was low-pass filtered). (f) Average rate vs. G for inputs in which the stimulus was low-pass filtered at140 Hz and the noise was low-pass filtered at either 20 Hz or 1000 Hz. S/N¼ 10. Values werecalculated by considering 25 experiments of 50 trials each. Stimuli were six seconds induration, with the last five seconds used in analysis. Error bars (SEs) are smaller than thesymbols. Curves for 20 Hz and 1000 Hz are almost indistinguishable. (g) Average jitter vs. Gfor same data set. Error bars are smaller than symbols.


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frequencies that were heavily filtered by the neuron. For 20 Hz noise, the jitter

decreased by a factor of approximately 3 when G increased from 0.1 to 0.8 mS. For

1000 Hz noise, the effect was considerably smaller, with jitter decreasing by a factor

of about 1.8 and with little variation (<10%) observed for conductances �0.4 mS.

Jitter calculations when the S/N was decreased to 5:1 or increased to 40:1 (by

changing the amplitude of the random waveform) produced qualitatively similar

results (data not shown). The jitter increased or decreased overall, respectively, as

expected, but the dependence on conductance for each of the two noise frequencies

was very similar to that seen in Figure 1(g) for both S/N ratios.

Modification of rate and jitter in HH8 model through changes in maximal

conductances

In the simulations using the LIF model, jitter was largely independent of

conductance for G� 0.4 mS when the bandwidth of the random waveform was

1000 Hz. We wondered if, in a more biophysically realistic model, jitter resulting

from high frequency noise might be modified through changes in the maximal

conductances of specific voltage-dependent ion channels via a mechanism more

complex than simple RC filtering. We furthermore postulated that in such a model

the relationship between spike rate and jitter would be more complicated, or at least

different, than in the LIF neuron.

To explore this issue, we employed a set of 9946 tonically-spiking parameter-

izations of an eight-channel Hodgkin Huxley type (HH8) model neuron chosen at

random from a previously published database (Prinz et al. 2003a, 2003b). For each

set of parameters, the model was injected with a current stimulus that consisted of a

0.78 nA rms signal low pass-filtered at 140 Hz to which GWN was added with a S/N

of 40:1. Figure 2(a) summarizes the results of the simulations and plots mean jitter

versus mean spike rate for each version of the model tested. Slow, narrow spikers

(1942 occurrences) and fast spikers (8004 occurrences) had distinct, largely non-

overlapping behaviors and are shown in black and gray, respectively. Nearly all the

slow spikers were found in a strip, tilted a bit to the right of vertical, at frequencies

below 10 Hz and with jitters ranging between approximately 0.15 and 2 ms with a

mean value of 0.72 ms. For slow spikers, rate and jitter were positively correlated

(correlation coefficient CC¼ 0.53, P¼ 0). The vast majority of the fast spikers were

found at frequencies above 25 Hz. Their jitters were typically below 1 ms

(mean¼ 0.25 ms) and tended to decrease with increasing rate; here the correlation

between rate and jitter was negative (CC¼�0.60, P¼ 0).

The inset of Figure 2(a) shows an expanded view of the main plot between 15 and

16 Hz where there were very few data points but where both slow and fast spikers

could be found. The black stars represent three model neurons, all slow spikers, that

fired at nearly the same rate (15.2–15.4 Hz) but that exhibited very different values

of jitter. Voltages responses, raster plots and PSTHs for the three sets of parameters

are shown at the left of Figure 2(b) (jitter¼ 0.51 ms), Figure 2(c) (1.12 ms) and

Figure 2(d) (1.76 ms). At the right side of each of Figure 2(b)–(d) is a histogram

showing the timing of all spikes in all events referred to the mean event time in each

experiment and a bar chart indicating the values of the maximal conductances for

each of the eight channels in that particular parameterization.


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Figure 2. Rate and jitter are not simply related to maximal conductances or each other in theHodgkin-Huxley type 8 channel (HH8) model database. We studied 9946 parameterizationsof the model randomly chosen from the 263 114 parameterizations in the database that firedtonically in response to 0, 3 and 6 nA constant current injection. Each parameterization was

(continued)


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Although they exhibited very different levels of jitter, the three variations of the

model highlighted in Figure 2 shared some common features. For example, in each

case �gCaT ¼ 0, �gCaS ¼ 2 and �gA¼ 0, where the maximal conductances are reported as

multiples of the canonical values. Maximal conductances for other channel types, on

the other hand, varied considerably; �gH ranged from 0 to 5 and �gL from 1 to 5.

We wondered which conductance or conductances were responsible for the range of

jitter observed and whether any findings could be generalized. To address this

question, we calculated correlation coefficients between the maximal conductance

value and the spike rate or jitter for each of the eight channel types. We considered

slow and fast spikers separately; results of the correlation analysis are plotted in

Figure 3. Some channel types (H, Kd) seemed to have little influence on either rate

or jitter. For the channel types that did matter, the relationships between maximal

conductance and rate or jitter appeared to be more complicated than in the LIF

model, where rate and jitter both decreased with increased leak conductance. This

can be seen, for example, by considering KCa. For slow spikers, the correlation

coefficients were both negative, while for fast spikers the coefficients were larger in

magnitude and had different signs.

The correlation coefficients in Figure 3 were calculated using the entire set of

9946 parameterizations. Because these were chosen randomly from the population

of tonic spikers and not from the general population, it was not surprising that the

distributions of maximal conductance for some of the eight channel types were not

uniform for either slow or fast spikers. For example, although the distribution of

KCa maximal conductances for slow spikers was nearly uniform, varying from 311

parameterizations with �gKCa¼ 5 to 333 for �gKCa¼ 0 and 4, the distribution was very

skewed for fast spikers (�gKCa¼ 0 in 5754 instances, �gKCa¼ 1 in 1927 instances,

�gKCa¼ 2 in 312 instances, �gKCa¼ 3 in 11 instances). Some deviations from uniform

(continued)injected with a 6 second duration deterministic ‘‘signal’’ (white noise low-pass filtered at140 Hz, 0.78 nA rms) to which a random waveform (white noise filtered at 2000 Hz due tosampling) was added at a 40:1 S/N ratio. The stimulus was presented 200 times, and the jitterwas calculated by averaging over all events. (a) Average jitter versus spike rate for eachparameterization. Black dots and gray dots correspond to ‘‘slow, narrow spikers’’ and ‘‘fastspikers’’ from Hong et al. (2008). Inset: expanded view of the plot for a frequency rangewhere there were very few data points. The black stars correspond to three slow spikers thathad essentially the same spike rate but very different magnitudes of jitter. (b) Left side top:portion of current stimulus injected into each of these three parameterizations. Below thestimulus is the voltage trace corresponding to the response for the starred parameterizationwith the smallest jitter (0.51 ms) shown in the inset of (a) and rasters showing the spike timesfor 40 presentations, with the top raster corresponding to the voltage trace shown. Below therasters is a PSTH (bin size 0.5 ms) summarizing the responses to 200 presentations (‘‘trials’’);the dots under the PSTH indicate ‘‘events’’. Right side top: histogram showing the timing ofall spikes in all events referred to the mean event time. Below this is a bar chart indicating thevalues of the maximal conductances for each of the eight channels in this version of themodel. Maximal conductances are reported as multiples of the canonical values. (c) Left:voltage trace, rasters, PSTH for starred parameterization with intermediate jitter (1.12 ms)from inset of (a). Right: histogram showing the timing of all spikes in all events and bar chartindicating the values of the maximal conductances for this parameterization. (d) Left: voltagetrace, rasters, PSTH for starred parameterization with largest jitter (1.76 ms) from inset of(a). Right: histogram showing the timing of all spikes in all events and bar chart indicating thevalues of the maximal conductances for this parameterization.


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distributions were not unexpected; for example, no parameterization had a sodium

conductance equal to zero.

To explore whether the unequal distribution of conductance values affected the

analysis, we recalculated the correlation coefficients while forcing the number of

parameterizations with each conductance value to be the same. In the case of KCa/

slow spikers, we randomly chose 311 of these for each conductance value and

calculated the correlation coefficients. We repeated the process twice more

(randomly choosing different parameter sets), averaged the three results, and

obtained correlation coefficients of �0.17 (rate) and �0.049 (jitter). As expected,

these were very close to the values of �0.17 and �0.051 obtained using the entire

data set. For fast spikers there was a larger discrepancy; we calculated coefficients of

�0.35 (rate) and 0.24 (jitter) using the entire data set and �0.51 (rate) and 0.35

(jitter) using a uniform distribution. The latter result, in which the correlation

coefficients differed depending on whether the entire data set or a uniform

conductance distribution was analyzed, was more the exception than the rule. Only

Figure 3. Spike rate and jitter show both positive and negative correlations with maximalconductances for the HH8 model database. (a) The Pearson product-moment correlationcoefficient between spike rate and maximal conductance for slow (black) and fast spikers(gray) for each of the eight channel types. (b) Pearson correlation coefficient between jitterand maximal conductance. Almost all of the correlation coefficients were statisticallysignificant (�P < 0.05, ��P < 0.01, ��P < 0.001). No correlation analysis was done for the CaTmaximal conductance for slow spikers (#), because all except two of the parameterizationshad a CaT conductance of zero. Correlation coefficients were calculated using the entire set:1942 slow spikers and 8004 fast spikers. In a few cases, we found that using the entire data setrather than one with a uniform distribution of maximal conductance values resulted in asubstantial underestimation of the correlations. See text for details.


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in one other case (Kd in slow spikers) did we find that the correlation coefficients

were both appreciable (magnitude �0.1) and differed by more than 15% when

calculated via the two methods, and in that case the magnitude of both correlation

coefficients also increased when we required the maximal conductance distribution

to be uniform. Hence, using the entire data set appears to have resulted in, if

anything, an underestimation of the strengths of the correlations.

A correlation analysis can be used to determine the existence and strength of the

mathematical relationship between two variables in a data set; however, since it

reduces that relationship to a single number it reveals nothing of the underlying

structure in the data set that produces the correlation. To address this issue, we

more closely examined four of the conductances (Na, CaS, A, KCa) with the biggest

correlation coefficients for jitter. Here we focused on one channel type at a time,

determining how its maximal conductance affected rate and jitter, independent of

the maximal conductances of the other channels. Our results, shown in Figure 4, are

consistent with the correlation coefficients of Figure 3. Plots of rate or jitter versus

conductance value with steeper slopes corresponded to correlation coefficients with

larger magnitudes.

As discussed above, the maximal conductances of the 9,946 parameterizations of

the HH8 model considered were not randomly distributed. In a simpler but related

five-conductance model, Goldman et al. (2001) showed that spiking, bursting and

quiescent behavior occupy different distinct regions of maximal conductance space.

In our study, by restricting the analysis to a certain type of spiking behavior, we

expected to introduce correlations, both positive and negative, between the various

pairs of maximal conductances in our data set (Hudson and Prinz 2010).

Furthermore, we expected that these correlations might be different for the slow

and fast spikers. When we looked into this question, we found this, in fact, to be the

case. For example, although the KCa maximal conductance was not highly

correlated with any of the other seven conductances for slow spikers (magnitude of

CC < 0.06 for all pairs), for fast spikers the correlations with some conductances

were quite large: Na: 0.18, CaT: �0.24, CaS: 0.35, A: �0.12, Kd: 0.21. As such, we

wondered what conclusions could be safely drawn from the results shown in Figures

3 and 4. Did the correlations between jitter and maximal conductance indicate that

jitter could independently be modified by changing one individual channel type, or

were they the result of correlated changes involving multiple channel types?

To investigate this issue we considered pairs of parameterizations in the data set

for which, by chance, seven conductances were identical and the eighth differed by

only one unit. In Figure 5 we plot the jitters of these pairs for the four channels (Na,

CaS, A, KCa) on which we have focused. The diagonal line has slope one. For

points above the line, the jitter increased with the addition of one unit of

conductance for the particular channel; for those below the line, jitter decreased.

For the most part, the correlation coefficients with jitter do seem to be explained by

single channel effects. For slow spikers, the points for Na (CC¼�0.24 in

Figure 3(b)) and A (CC¼�0.16) are largely below the line while those for CaS

(CC¼ 0.42) are almost all above. For KCa, which had a correlation coefficient

(CC¼�0.05) with a small magnitude, increasing the conductance appears to have

had little effect since nearly all points fell close to the line. The plots in Figure 5 for

the fast spikers are also largely consistent with the correlation analysis, although the

results show more variability. The correlations coefficients for Na and CaS, for


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example, were small in magnitude, but for individual model pairs, modifying the

maximal conductance by one unit could result in huge changes in jitter. For these

two channel types, the small correlation coefficients reflected an inconsistent effect,

rather than no effect, as had been observed with KCa for the slow spikers.

Figure 4. Average spike rate and jitter depend on the maximal conductances of particularchannels. The average rate and jitter for slow spikers (left) and fast spikers (right) are plottedas a function of the maximal conductance of each of four channel types (Na, CaS, A, KCa).Maximal conductances are reported as multiples of the canonical values. Error bars (SEs) aresmaller than the symbols for all except one of the data points.


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Figure 5. Modifying the maximal conductance of one channel type by one unit can changejitter appreciably. The jitters of pairs of parameterizations that differed by only one unit inconductance for a particular channel (Na, CaS, A, KCa) are plotted. (The other sevenmaximal conductances were identical for the pair.) For each data point, the horizontalposition corresponds to the jitter for one member of a pair; the vertical position correspondsto the jitter for the other member of the pair in which the maximal conductance of theparticular channel type was one unit larger. The diagonal line has slope equal to one. Tofacilitate interpretation, the correlation coefficients between maximal conductance and jitterfrom Figure 3(b) have been printed on the graphs.


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Discussion

Mechanism for modification of jitter in LIF model

Although neurons are highly nonlinear, at voltages below the threshold for action

potential generation the electrical response is much less so and can be reasonably

approximated to first order by the LIF model. For current with amplitude I and

frequency f in such a circuit, the amplitude of the change in membrane potential V

(with respect to equilibrium) is given by

V ¼IRffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ 2�fRCð Þ2

p ð3Þ

At low frequencies, the membrane is always completely charged, all current

passes through the channels, and voltage is proportional to membrane resistance. At

frequencies well above the cutoff frequency f0¼ 1/(2�RC), the capacitance looks

approximately like a short circuit, and V is independent of R and drops as 1/f. In

general, an increase in conductance G, or equivalently a decrease in R, has two

effects: the magnitude of V is decreased for all frequencies, and f0 is pushed to

higher frequencies. In the LIF model, with its hard spiking threshold, these effects

result in reduced spiking overall, but with an expanded range of input frequencies

that can elicit spiking. The increase in f0 is equivalent to a decrease in the response

time constant �¼RC; with increased conductance, the shape of the waveform of the

analog portion of the voltage response contains higher frequencies and tracks the

input more faithfully. The PSDs shown in Figure 1(e) are consistent with this

description. For both values of conductance the curves were relatively flat at low

frequencies (below 10 Hz), decreased above the respective cutoff frequencies, and

then dropped rapidly above �140 Hz due to the fact that the deterministic part of

the input itself was low-pass filtered.

The subthreshold behavior of an LIF model subject to additive Gaussian noise is

an example of an Ornstein-Uhlenbeck process (Uhlenbeck and Ornstein 1930;

Gluss 1967). Jitter in the LIF model corresponds to variability in the ‘‘first passage’’

times of threshold crossings in an Ornstein-Uhlenbeck process, and this mathe-

matical problem has been analyzed in considerable detail (Ricciardi and Sacerdote

1979; Wan and Tuckwell 1982; Tuckwell and Wan 1984). Simply put, however, the

relative importance of noise in determining jitter depends on the slope of the voltage

when it crosses threshold. The magnitude of the jitter Dt corresponds roughly to the

time interval during which the sum of the deterministic and random parts of the

voltage waveform could sum to Vthresh. Over a few milliseconds, the deterministic

part of the voltage signal for our LIF neuron varied approximately linearly with time,

and jitter can be related to the voltage waveform through the following equation:

dV0

dt

��Dt � Vnoise ð4Þ

where dV0/dt is the slope of the deterministic part of the voltage signal V0, and Vnoise

is the amplitude of the voltage noise that resulted from the random current

waveform. The jitter can thus be approximated by the following:

Dt �Vnoise

dV0=dt�� ð5Þ


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In the case of a simple hyperpolarization of the neuron through constant current

injection (Figure 1c), neither the numerator nor the denominator of Equation 5

changes, on average. Although the jitter of particular events might increase or

decrease, no average change in jitter would be expected, and this is what we

observed. What did change was the timing of events. As seen in Figure 1(d), events

occurred later when the neuron was hyperpolarized (bottom panel) than under the

original conditions (top). This can be understood by comparing the voltage

waveforms in Figure 1(a) and (c). They are largely identical, except for a voltage

offset. Spiking was delayed in the latter case, since thresholds were reached slightly

later, higher up on the local upswings in voltage.

When conductance was increased fourfold (Figure 1b), jitter decreased and the

timing of events changed. The effect on jitter could largely be explained in terms of

the changes in the membrane time constant. Increased conductance reduced the

time constant and sharpened features in the voltage waveform, as can be seen by

comparing the voltage waveforms in Figure 1(a) and (b). Larger slopes in the voltage

trajectory as the neuron approached threshold (i.e. a larger denominator in

Equation 5) resulted in reduced jitter. The effect on timing was more complicated

than in the case of hyperpolarization, as can be seen in the middle panel of

Figure 1(d), where the first event preceded and the second event lagged the

corresponding events in the top panel. Here, it appears two competing factors were

at play. The reduction of the voltage waveform amplitude (Figure 1b) induced lags

in the event times, similar to the effect of hyperpolarization; however, the decrease in

the time constant sharpened the features in the waveform and pushed events

forward in time.

For the S/N ratios investigated, the dependence of the rate on conductance was

identical for the random waveforms filtered at 20 Hz and 1 kHz (Figure 1f). The

shapes of the graphs of jitter versus conductance (Figure 1g) differed, however, in that

jitter decreased over the entire range of conductance for the 20 Hz bandwidth,

whereas the plot flattened out halfway through the range when the random waveform

was filtered at 1 kHz. To understand the flattening of these curves required us to

consider the effects of changing conductance on both the numerator and denom-

inator of Equation 5. How the numerator and denominator are affected by changes in

G depends on how the frequencies in the deterministic and random parts of the

current input compare to the cutoff frequency. Consider a narrow frequency

spectrum associated with the random part of the input. If these frequencies were all

well below f0 (i.e. in the ohmic frequency-independent regime of Equation 3), then

Vnoise would be proportional to 1/G (i.e. proportional to R). If instead the frequencies

were all far above f0, then Vnoise would be independent of G but would be proportional

to 1/f. The relationship between V0 and G would depend on the frequencies

associated with the deterministic part of the signal in a similar fashion: proportional to

1/G for frequencies below f0 and independent of G but proportional to 1/f for

frequencies above f0. The time-derivative results in an additional factor of frequency f

in the denominator of Equation 5 but does not affect the dependence on G.

In our simulations the frequency bands were broad and often straddled the cutoff

frequency, making the analysis more difficult. First, consider the case of 20 Hz

noise. An increase in the conductance from 0.1 to 0.8 mS corresponded to an

increase in the cutoff frequency f0 from 16 to 127 Hz. For lower values of

conductance, roughly speaking, the noise (i.e. random waveform) was slower than f0


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(numerator / 1/G) and most of the deterministic signal was faster than f0(denominator independent of G). Hence, the naı̈ve, narrow band calculation would

predict jitter to be proportional to 1/G. As conductance was increased, more and

more of the deterministic signal fell below f0, changing the behavior of the

denominator, and one would expect jitter to eventually become independent of G.

What was observed (Figure 1(g), filled symbols) was qualitatively consistent with

this description: jitter decreased over the entire range of conductance, although

more weakly than as 1/G, and the curve flattened out as conductance increased.

The case of 1 kHz was more complicated. For low values of G, most of the

deterministic signal and even more of the random waveform were above f0, and so

the ratio in Equation 5 should be independent of G. However, as conductance was

increased and additional frequencies contributed to both numerator and denom-

inator, the additional factor of f in the denominator should result in the ratio

decreasing, but in a weaker fashion than for the 20 Hz case. This is what was

observed (Figure 1(g), open symbols). Interestingly, if the signal and noise spectra

were chosen such that we could increase G to the point where the entire

deterministic signal was below f0 and most of the noise was above f0, the jitter versus

conductance curve would be predicted to be roughly proportional to G. Because the

slope would have changed from negative to positive, there would be a conductance

for which the jitter was minimized. At even larger values of G, all frequencies would

be below the cutoff, and the slope would be expected to return to zero.

As we had initially speculated, the neuromodulatory effects that had been

previously seen in CBCTO responses can be explained, at least qualitatively, in

terms of changes in the filtering associated with passive membrane properties.

Increased conductance in the presence of the peptide allatostatin decreased both

spike rate and jitter in CBCTO. Serotonin, which decreased conductance, had the

opposite effects on rate and jitter (Billimoria et al. 2006).

Analysis of factors influencing rate and jitter for parameterizations of the HH8 model

Conductances in the HH8 model (and in real neurons) are both voltage- and time-

dependent, and we expected the connection between spike rate and jitter to be more

complicated than in the LIF model. Figure 2 shows that for a particular jitter there

can be a wide range of spike rates and vice versa. Although complex, the relationship

between rate and jitter in the HH8 model was not without structure. In particular,

the two were positively correlated for slow spikers and negatively correlated for fast

spikers.

The simplest approach to understanding the relationship between maximal

conductances and spike rate considers the role of each channel type independently:

maximal conductances associated with inward currents should be positively

correlated with spike rate while those associated with outward currents should be

negatively correlated. In our simulations using the HH8 model, this held true for a

number of the channel types: the two calcium conductances were positively

correlated with spike rate, and the transient potassium (A) and KCa conductances

were negatively correlated (Figure 3). In a real neuron or in a conductance-based

model, the conductances are not independent, and one must consider the functional

interactions between them, for instance, through the membrane potential or Ca2þ

concentration. We would not have been too surprised, for example, if the correlation


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between rate and a maximal calcium conductance had been negative, because

increased calcium influx also increases the magnitude of IKCa, an outward current.

By restricting our analysis to tonic spikers, we introduced correlations between

some pairs of maximal conductances that, in a sense, allowed the associated

channels to interact with each other, at least in our correlation analysis. Such an

interaction likely explains the differences in the correlations observed between the

sodium maximal conductance and spike rate for slow and fast spikers. For both

types of spikers the sodium current is zero by �5 ms after a spike (Hong et al. 2008)

and does not begin again until the next spike. Hence an increase or decrease in the

sodium maximal conductance should not have much effect on spike rate, as long as

there is enough sodium to produce spikes in the first place, and naı̈vely one would

not expect to see large correlations between conductance and rate for either type of

spiker. Very little was found for slow spikers (CC¼ 0.056), but a large negative

correlation (CC¼�0.41) was observed for fast spikers. This latter effect likely was

due, not to a direct or indirect effect of the Na current, but to the positive

correlation between the Na and KCa maximal conductances (CC¼ 0.18), the latter

of which is associated with an outward current.

The other striking feature in the analysis of rate and maximal conductance, the

large positive correlation between the spike rate and the leak conductance for slow

spikers (Figure 3a), is easier to explain. After spiking, slow spikers spend

considerably more time below the leak reversal potential (�50 mV) (Prinz et al.

2003a) than do fast spikers (80–100 vs. 5–15 ms) (Hong et al. 2008). Leak thus is a

key depolarizing current for slow spikers but does not play an important role in the

excitability of the fast spikers.

In addition to having different spike rates, slow and fast spikers exhibited very

different temporal precision in their spike trains. The mean jitter for slow spikers

(0.72 ms) was nearly three times that for fast spikers (0.25 ms). This can be

understood in the context of Equation 5. Fast spikers activate their depolarizing

membrane conductances much more quickly after a spike than slow spikers (Hong

et al. 2008). During the upstroke toward the next spike in a fast spiker, any current

noise would compete with a much higher membrane conductance and more

rapidly-varying membrane potential than in the case of slow spikers, and, as a result,

noise would have less influence on the exact timing of the spike.

Uncovering the roles of the individual currents in determining jitter is more

challenging. Knowledge of the correlations between maximal conductance and rate

in the HH8 model (Figure 3a) does not allow prediction of even the sign of the

correlations between conductance and jitter (Figure 3b). The plots in Figure 5 in

which nearly all the points are above or below the line suggest that, at least in some

cases, effects on jitter can be caused by the change in the maximal value of a single

conductance. In those instances, some intuition can be obtained by consideration of

the associated membrane current before, during, and after spiking. For example,

Hong et al. (2008) showed that in spontaneously spiking neurons, the dominant

behavior of IA for both slow and fast spikers is largely the same: it increases rapidly

upon spiking. A small difference in the behaviors, however, likely explains the

opposite signs in the correlation coefficients between jitter and maximal conduc-

tance. For slow spikers, IA is zero prior to the spike, and the increase in the current is

tightly timed (<1 ms) with respect to the spike’s voltage peak. An increase in the

maximal conductance would be expected to decrease jitter, which is what was


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observed in our simulations (Figures 3–5). For fast spikers, the increase in current

was also precisely timed; however, before each spike IA was non-zero and nearly

constant in magnitude. Hence, prior to spiking, IA contributed to a flattening of the

membrane potential, making the neuron more susceptible to noise and increasing

jitter (Equation 5).

Explanations for the correlations between jitter and the maximal conductances for

the Na and Ca channels are in the same vein. In spontaneously active slow spikers,

the onset of INa is very precisely timed (Hong et al. 2008), and, in our study, the

correlation coefficient between maximal conductance and jitter was negative

(CC¼�0.24). For fast spikers an increase in maximal conductance was weakly

correlated with increased jitter (CC¼ 0.11), although the paired data in Figure 5

show that there was no consistent effect. The small positive correlation coefficient

presumably resulted from a less precisely timed INa (Hong et al. 2008) and/or from

the positive correlation between maximal conductances for Na and KCa, the latter

of which itself is positively correlated with jitter. An increase in either

calcium maximal conductance resulted in decreased jitter in fast spikers, on the

whole (Figures 3–5). In fast spikers, both calcium currents are essentially always

‘‘on,’’ and the direction of the current switches nearly instantaneously precisely at

the spike time (Hong et al. 2008), which might allow for a faster depolarization and

less susceptibility to noise. In contrast, the CaS maximal conductance had a strong,

positive correlation with jitter (CC¼ 0.42) in slow spikers, but a direct mechanism is

hard to identify. Although a change of one unit in CaS conductance had a huge

effect on jitter (Figure 5), ICaS does not appear to turn on until after the spike is

underway and has decayed back to zero before the next spike is produced

(Hong et al. 2008).

4.3 Comparisons with other studies and conclusions

To date there have been few studies of the roles played by individual channel types

in determining the temporal precision of spike trains. One of the best studied

systems involves vertebrate pyramidal cells, where it has been demonstrated that the

persistent sodium current INaP (which is not included in our HH8 model) can both

enhance and reduce spiking precision, and that which of these effects is observed

depends on the stimulus protocol. Vervaeke et al. (2006) showed that INaP increased

the regularity of spiking in response to constant depolarizing current injection by

amplifying afterhyperpolarizations (AHPs), which themselves have been credited

with improving the temporal precision of spike trains (de Ruyter van Steveninck

et al. 1997; Berry and Meister 1998). In other experiments it was shown that when

EPSPs were evoked by axonal stimulation, INaP prolonged their duration and

reduced spike time precision (Andreasen and Lambert 1999; Fricker and Miles

2000; Zsiros and Hestrin 2005). The temporal variability of spiking in these

experiments was large (tens of ms) and only a single spike was evoked in each trial.

Nonetheless, this second result is relevant to our own study. For the prolonged

EPSPs, the denominator of Equation 5 becomes very small in the vicinity of the

broadened peak. In the presence of INaP, as in the case of our slow spikers, the

membrane potential spends more time in the vicinity of threshold during which it is

sensitive to stochastic noise that can trigger imprecise spiking (Vervaeke et al. 2006).


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Computational studies using models simpler than the HH8 have yielded insights

into the roles of specific channel types in influencing jitter. Gutkin et al. (2003)

showed that spiking precision and reliability results from experimental investigations

could be reproduced using a �-neuron, a canonical representation of neurons with

Type I excitability. This model, which is not conductance-based, does not account

for slow currents, including those that are responsible for spike-rate adaptation. In a

six-conductance, single-compartment model of cortical neurons, Schreiber et al.

(2004) found that variations in all of the conductances could shift the cell’s

‘‘reliability measure’’ (Schreiber et al. 2003) in response to inputs consisting of a

sinusoidal signal to which DC and filtered Gaussian noise were added. For the most

part, these shifts could be explained by changes in the cell’s spike rate and hence its

preferred frequency, as per the resonance effect mentioned earlier. Slow potassium

channels, however, were found to improve spike timing beyond shifting the cell’s

resonant frequency, and the authors suggested that increasing the maximal

conductance reduced the sensitivity to noise after a spike by deepening the AHP.

More recently, Prescott and Sejnowski (2008) showed in a modified five-

conductance Morris-Lecar model that two channels that have similar effects on

average spike rate can have very different ones on spike-time precision.

The presence of either the voltage-activated M-type Kþ current (IM) or the

calcium-activated Kþ current (IAHP) increased spike rate adaptation and hence

reduced the spike rate. However, in response to a time-varying signal (similar to that

used in our study) in the presence of noise, spike timing was disrupted and jitter

enhanced in the presence of IAHP, but was minimally affected by IM.

In this last study, the authors were able to gain insight into the mechanisms by

which IM and IAHP influenced spike timing through considerations of nullclines and

bifurcation analysis. The higher dimensionality of the HH8 model unfortunately

precludes a straightforward application of these techniques or a simple analysis of

the roles of the various conductances. The behaviors we observed in our study were

complicated, but the neurons that inspired the HH8 model (Turrigiano et al. 1995)

themselves are complicated and exhibit a wide and rich variety of behaviors. The

complexity of the HH8 model can be reduced through approximations, but without

detailed study it is unclear what aspects of the response are lost and over what range

of inputs the approximations are useful. Still, mathematical modeling has provided a

valuable tool for us to gain some general understanding of the relationships between

spiking and jitter. We have observed, first, that the dependence of jitter on

conductance in the LIF model can be largely explained in terms of RC filtering. In

the full-blown HH8 model, standard RC filtering can explain some but not all of

what is observed. Second, a channel’s voltage-dependent properties and kinetics,

particularly at or near the threshold for spiking, appear to play a major role in

determining temporal precision in spike trains. Third, jitter likely will show different

correlations with spike rate and with maximal conductances in situations in which

the interspike interval is smaller or comparable to transition times between states of

channel activity and inactivity versus when it is much longer. Fourth, several of the

correlation coefficients between jitter and maximal conductance in our study had

opposite signs for slow versus fast spikers, and hence the role of a particular channel

type in determining jitter is not uniquely defined but is dependent upon the

maximal conductances of the other channel types that are present. In our study we

considered only a single, rather arbitrary stimulus. We would expect that the relative


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importance of the various conductances and even the polarity of the effects they

have on jitter could be modified when the amplitudes (AC and DC) and frequency

spectrum of the stimulus are changed. In the future it will be interesting to extend

this work to determine the effect of modifying maximal conductances on other

properties of the resulting spike trains, such as reliability and information rate.

Acknowledgements

We thank E. Marder, M. Goldman and F. Chance for their helpful suggestions and

observations.

Declaration of interest: The authors report no conflicts of interest. The authors

alone are responsible for the content and writing of the paper. This work was

supported by a Provost Research Fellowship from Santa Clara University (VQZ).

References

Andreasen M, Lambert JD. 1999. Somatic amplification of distally generated subthreshold EPSPs in rat

hippocampal pyramidal neurones. Journal of Physiology 519.1:85–100.

Beierholm U, Nielsen CD, Ryge J, Alstrom P, Kiehn O. 2001. Characterization of reliability of spike

timing in spinal interneurons during oscillating inputs. Journal of Neurophysiology 86:1858–1868.

Berger T, Levy WB. 2010. A mathematical theory of energy efficient neural computation and

communication. IEEE Transactions on Information Theory 56:852–874.

Berry 2nd MJ, Meister M. 1998. Refractoriness and neural precision. Journal of Neuroscience

18:2200–2211.

Billimoria CP, DiCaprio RA, Birmingham JT, Abbott LF, Marder E. 2006. Neuromodulation of spike-

timing precision in sensory neurons. Journal of Neuroscience 26:5910–5919.

Bryant HL, Segundo JP. 1976. Spike initiation by transmembrane current: a white-noise analysis. Journal

of Physiology 260:279–314.

Butts DA, Weng C, Jin J, Yeh CI, Lesica NA, Alonso JM, Stanley GB. 2007. Temporal precision in the

neural code and the timescales of natural vision. Nature 449:92–95.

Carr CE, Konishi M. 1990. A circuit for detection of interaural time differences in the brain stem of the

barn owl. Journal of Neuroscience 10:3227–3246.

Crowe G, Rupert AL, Moushegian G. 1978. Phase locking in monoaural and binaural medullary

neurons: implications for binaural phenomena. Journal of the Acoustical Society of America

64:493–501.

Cull-Candy SG, Usowicz MM. 1989. Whole-cell current noise produced by excitatory and inhibitory

amino acids in large cerebellar neurones of the rat. Journal of Physiology 415:533–553.

de Ruyter van Steveninck RR, Lewen GD, Strong SP, Koberle R, Bialek W. 1997. Reproducibility and

variability in neural spike trains. Science 275:1805–1808.

DeFelice LJ. 1981. Introduction to membrane noise. New York: Plenum Press.

DeFelice LJ, Wanke E, Conti F. 1975. Potassium and sodium current noise from squid axon

membranes. Federation Proceedings 34:1338–1342.

Diba K, Lester HA, Koch C. 2004. Intrinsic noise in cultured hippocampal neurons: Experiment and

modeling. Journal of Neuroscience 24:9723–9733.

DiCaprio RA, Billimoria CP, Ludwar B. 2007. Information rate and spike-timing precision of

proprioceptive afferents. Journal of Neurophysiology 98:1706–1717.

Fricker D, Miles R. 2000. EPSP amplification and the precision of spike timing in hippocampal neurons.

Neuron 28:559–569.

Gerstner W, Naud R. 2009. Neuroscience. How good are neuron models? Science 326:379–380.

Gerstner W, van Hemmen JL, Cowan JD. 1996. What matters in neuronal locking? Neural Computation

8:1653–1676.


Net

wor

k D

ownl

oade

d fr

om in

form

ahea

lthca

re.c

om b

y E

mor

y U

nive

rsity

on

01/1

9/15

For

pers

onal

use

onl

y.

Gluss B. 1967. A model for neuron firing with exponential decay of potential resulting in diffusion

equations for probability density. The Bulletin of Mathematical Biophysics 29:233–243.

Goldberg JM, Brownell WE. 1973. Discharge characteristics of neurons in anteroventral and dorsal

cochlear nuclei of cat. Brain Research 64:35–54.

Goldberg JM, Smith CE, Fernandez C. 1984. Relation between discharge regularity and responses to

externally applied galvanic currents in vestibular nerve afferents of the squirrel monkey. Journal of

Neurophysiology 51:1236–1256.

Goldman MS, Golowasch J, Marder E, Abbott LF. 2001. Global structure, robustness, and modulation

of neuronal models. Journal of Neuroscience 21:5229–5238.

Gutkin B, Ermentrout GB, Rudolph M. 2003. Spike generating dynamics and the conditions for spike-

time precision in cortical neurons. Journal of Computational Neuroscience 15:91–103.

Haas JS, White JA. 2002. Frequency selectivity of layer II stellate cells in the medial entorhinal cortex.

Journal of Neurophysiology 88:2422–2429.

Heiligenberg W. 1991. Neural nets in electric fish. Cambridge: MIT Press.

Hong E, Gurel Kazanci F, Prinz AA. 2008. Different roles of related currents in fast and slow spiking of

model neurons from two phyla. Journal of Neurophysiology 100:2048–2061.

Hudson AE, Prinz AA. 2010. Conductance ratios and cellular identity. PLOS Computational Biology

6:e1000838.

Hunter JD, Milton JG. 2003. Amplitude and frequency dependence of spike timing: implications for

dynamic regulation. Journal of Neurophysiology 90:387–394.

Hunter JD, Milton JG, Thomas PJ, Cowan JD. 1998. Resonance effect for neural spike time reliability.

Journal of Neurophysiology 80:1427–1438.

Izhikevich EM. 2004. Which model to use for cortical spiking neurons? IEEE Transactions on Neural

Networks 15:1063–1070.

Jacobson GA, Diba K, Yaron-Jakoubovitch A, Oz Y, Koch C, Segev I, Yarom Y. 2005. Subthreshold

voltage noise of rat neocortical pyramidal neurones. Journal of Physiology 564.1:145–60.

Kobayashi R, Tsubo Y, Shinomoto S. 2009. Made-to-order spiking neuron model equipped with a

multi-timescale adaptive threshold. Frontiers in Computational Neuroscience 3:9.

Kuwabara N, Suga N. 1993. Delay lines and amplitude selectivity are created in subthalamic auditory

nuclei: the brachium of the inferior colliculus of the mustached bat. Journal of Neurophysiology

69:1713–1724.

Mackevicius EL, Best MD, Saal HP, Bensmaia SJ. 2012. Millisecond precision spike timing shapes

tactile perception. Journal of Neuroscience 32:15309–15317.

Mainen ZF, Sejnowski TJ. 1995. Reliability of spike timing in neocortical neurons. Science

268:1503–1506.

Nemenman I, Lewen GD, Bialek W, de Ruyter van Steveninck RR. 2008. Neural coding of natural

stimuli: information at sub-millisecond resolution. PLOS Computational Biology 4:e1000025.

Ozer M, Uzuntarla M, Perc M, Graham LJ. 2009. Spike latency and jitter of neuronal membrane patches

with stochastic Hodgkin-Huxley channels. Journal of Theoretical Biology 261:83–92.

Pillow JW, Paninski L, Uzzell VJ, Simoncelli EP, Chichilnisky EJ. 2005. Prediction and decoding of

retinal ganglion cell responses with a probabilistic spiking model. Journal of Neuroscience

25:11003–11013.

Prescott SA, Sejnowski TJ. 2008. Spike-rate coding and spike-time coding are affected oppositely by

different adaptation mechanisms. Journal of Neuroscience 28:13649–13661.

Prinz AA, Billimoria CP, Marder E. 2003a. Alternative to hand-tuning conductance-based models:

construction and analysis of databases of model neurons. Journal of Neurophysiology 90:3998–4015.

Prinz AA, Thirumalai V, Marder E. 2003b. The functional consequences of changes in the strength and

duration of synaptic inputs to oscillatory neurons. Journal of Neuroscience 23:943–954.

Reinagel P, Reid RC. 2000. Temporal coding of visual information in the thalamus. Journal of

Neuroscience 20:5392–5400.

Ricciardi LM, Sacerdote L. 1979. The Ornstein-Uhlenbeck process as a model for neuronal activity.

I. Mean and variance of the firing time. Biological Cybernetics 35:1–9.

Rokem A, Watzl S, Gollisch T, Stemmler M, Herz AV, Samengo I. 2006. Spike-timing precision

underlies the coding efficiency of auditory receptor neurons. Journal of Neurophysiology

95:2541–2552.

Rolls ET, Treves A. 2011. The neuronal encoding of information in the brain. Progress in Neurobiology

95:448–490.


Net

wor

k D

ownl

oade

d fr

om in

form

ahea

lthca

re.c

om b

y E

mor

y U

nive

rsity

on

01/1

9/15

For

pers

onal

use

onl

y.

Rowat PF, Elson RC. 2004. State-dependent effects of Na channel noise on neuronal burst generation.

Journal of Computational Neuroscience 16:87–112.

Schreiber S, Fellous JM, Tiesinga P, Sejnowski TJ. 2004. Influence of ionic conductances on spike

timing reliability of cortical neurons for suprathreshold rhythmic inputs. Journal of Neurophysiology

91:194–205.

Schreiber S, Fellous JM, Whitmer D, Tiesinga P, Sejnowski TJ. 2003. A new correlation-based measure

of spike timing reliability. Neurocomputing 52-54:925–931.

Stein RB. 1967. Some models of neuronal variability. Biophysical Journal 7:37–68.

Sullivan WE, Konishi M. 1984. Segregation of stimulus phase and intensity coding in the cochlear

nucleus of the barn owl. Journal of Neuroscience 4:1787–1799.

Swerup C, Rydqvist B. 1996. A mathematical model of the crustacean stretch receptor neuron.

Biomechanics of the receptor muscle, mechanosensitive ion channels, and macrotransducer

properties. Journal of Neurophysiology 76:2211–2220.

Tang AC, Bartels AM, Sejnowski TJ. 1997. Effects of cholinergic modulation on responses of neocortical

neurons to fluctuating input. Cerebral Cortex 7:502–509.

Toups JV, Fellous JM, Thomas PJ, Sejnowski TJ, Tiesinga PH. 2011. Finding the event structure of

neuronal spike trains. Neural Computation 23:2169–2208.

Tuckwell HC, Wan FYM. 1984. First-passage time of Markov processes to moving barriers. Journal of

Applied Probability 21:695–709.

Turrigiano GG, LeMasson G, Marder E. 1995. Selective regulation of current densities underlies

spontaneous changes in the activity of cultured neurons. Journal of Neuroscience 15:3640–3652.

Uhlenbeck GE, Ornstein LS. 1930. On the theory of Brownian Motion. Physical Review 36:823–841.

Vervaeke K, Hu H, Graham LJ, Storm JF. 2006. Contrasting effects of the persistent Naþ current on

neuronal excitability and spike timing. Neuron 49:257–270.

Wan FYM, Tuckwell HC. 1982. Neuronal firing and input variability. Journal of Theoretical

Neurobiology 1:197–218.

White JA, Klink R, Alonso A, Kay AR. 1998. Noise from voltage-gated ion channels may influence

neuronal dynamics in the entorhinal cortex. Journal of Neurophysiology 80:262–269.

Zsiros V, Hestrin S. 2005. Background synaptic conductance and precision of EPSP-spike coupling at

pyramidal cells. Journal of Neurophysiology 93:3248–3256.


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pers

onal

use

onl

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