Spontaneous action potentials and neural coding in

un-myelinated axons

Cian O’Donnell1,2 and Mark C.W. van Rossum1,⇤

September 23, 2014

1Institute for Adaptive and Neural Computation, School of Informatics, University ofEdinburgh, UK2Current address: Computational Neurobiology Laboratory, Salk Institute for BiologicalStudies, La Jolla CA, USA⇤Corresponding author: mvanross@inf.ed.ac.uk

Abstract

The voltage-gated Na and K channels in neurons are responsible for action poten-

tial generation. Because ion-channels open and close in a stochastic fashion, sponta-

neous (ectopic) action potentials can result even in the absence of stimulation. While

spontaneous action potentials have been studied in detail in single compartment mod-

els, studies on spatially extended processes have been limited. The simulations and

analysis presented here show that spontaneous rate in un-myelinated axon depends

non-monotonically on the length of the axon, that the spontaneous activity has sub-

Poisson statistics, and that neural coding can be hampered by the spontaneous spikes

by reducing the probability of transmitting the first spike in a train.

Introduction

Action potentials rely crucially on the voltage-gated channels in the neural membrane. Theclassic Hodgkin-Huxley model describes how small depolarizations of the membrane lead toincreased opening of the Na channels, the ion-flux through which further depolarizes themembrane. Once this process reaches a point of no return, an action potential is generatedwhich is terminated by the opening of the hyper-polarizing K channels that follow themembrane potential more sluggishly than the Na channels. As the opening of the channelsis a stochastic process, the action potentials have a stochastic character as well. This is mostnoticeable in the initiation phase of the spike when only few channels are open. During thelater phase, so many channels open that stochastic fluctuations are largely averaged out.

The stochastic character of ion channels has been studied extensively in both experimen-tal and modeling literature. A number of modeling studies has considered single compart-ment models, in which a cell is lumped together in one iso-potential sphere. For sphereswith small membrane areas, stochastic opening and closing of channels can lead to sponta-neous action potentials, even in the absence of any stimulation. For large membrane areas,the stochastic character disappears and the system is well described by the deterministicHodgkin-Huxley equations. In particular, the rate of spontaneous action potentials decreasesexponentially with the membrane area (see below and Chow and White, 1996).

Similarly, for un-myelinated axons the spontaneous action potential rate increases as thediameter of the axon thins. It has been argued using simulations that in 1mm long axonsthinner than ⇠ 0.1µm, spontaneous firing rates become so substantial that they severelyinterfere with the communication of stimulus-driven action potentials (Faisal, White, and

1

Laughlin, 2005). This effect might therefore impose a limit on the minimal diameter ofaxons, but length dependence has not been examined.

In this study we examine in simulations the relation the spontaneous rate and the axonlength. Interestingly, depending on diameter, the spontaneous rate is non-monotonic inlength, it is high for both short and long axons, but lowest for intermediate length axons.Our subsequent reduced model allows to explore these effects exactly and identifies variousregimes each with its particular characteristics. Next, we show how spontaneous spikesinterfere with information coding. The spontaneous spikes have a peculiar characteristicsdifferent from any other sources of noise studied before in neurons, that can hamper codingbut can also be used to identify spontaneous spiking activity experimentally.

Results

We consider a uniform, un-myelinated axon with sealed ends and use the standard Hodgkin-Huxley model to describe the active properties of the axon. The only difference in thestandard parameters is that we vary the diameter and length of the axon. While the HHmodel at 6.3C can hardly be called a realistic model for mammalian neurons, we use itbecause the full characterization of ion channels in mammalian un-myelinated axons is in-complete. Furthermore, using the HH model has the advantage that both its deterministicand stochastic variants have been studied extensively. The stochastic simulation were doneusing both the PSICS simulator (Cannon, O’Donnell, and Nolan, 2010), and the stochasticmodule in NEURON (Carnevale and Hines, 2006).

Rate vs length We first study the spontaneous rate in axons of arbitrary diameter andlength, Fig.1A. For a given diameter the spontaneous rate is non-monotonic in the length: itis high for very short and very long axons, but lower for intermediate lengths. This behaviorcan be understood as follows: When the axon is very long, it can be seen as a collection ofmany weakly coupled compartments, each of which can generate spikes. Yet once a spike isgenerated, it will travel along the whole axon (unless it collides with another spontaneousspike, see below). As the number of such compartments and thus the number of independentinitiation zones increases with axon length, the rate increases with length for long enoughaxons. However, for very long axons the rate saturates due to collisions.

In contrast, very short axons can be collapsed into a single compartment if they are muchshorter than the electrotonic length (or space-constant) ⇤. Even in the single compartmentcase the analysis of spontaneous spiking in the HH model is complicated (see Chow andWhite (1996), and corrections in Keener and Newby (2011); Newby, Bressloff, and Keener(2013); O’Donnell and Van Rossum (2014)). Fortunately the dependence of the spontaneousrate on area and thus length is well understood (Chow and White, 1996). In analogy witha Kramer’s escape problem the spontaneous firing rate f behaves as

f / exp(�c/�2V

) (1)

where �2V

is variance in the membrane voltage and c is some constant. Under the assumptionof small fluctuations, the variance in the voltage equals �2

V

=

´SI

(f)|Z(f)|2df , whereSI

(f) is the power-spectrum of the noise current, and Z(f) the membrane impedance.This voltage variance is in general a complicated expression, as the channel noise has acolored power-spectrum and the membrane impedance in the HH model has a resonance(O’Donnell and Van Rossum, 2014). However, here we only need the dependence on themembrane area. Assuming that the voltage is approximately constant, the channels areindependent and the spectral density of the fluctuations in the current are proportional tothe number of channels and hence proportional to the membrane area S

I

(f) / A. However,the impedance is inversely proportional to area, Z(f) / 1/A. So that we find that �2

V

/ 1/Aand log f / �A / �L. This exponential dependence on length for short axons together with

2

the behavior for long axons determines the shape of Fig.1A: the spontaneous rate increasesfor both very short and very long axons.

Effect of diameter and spike initiation zones While spikes can be generated anywherein the axon, Fig.1D, the initiation is not uniform. A histogram of the initiation locationshows that most spikes are generated at the ends of the axon, Fig.1B. This can be under-stood using an argument similar to the one above explaining the area dependence in singlecompartments. The variance of the noise currents scales with the number of channels inthe local neighborhood. There are twice as many neighboring channels in the middle of theaxon as there are at the end. However, the impedance at the end is twice that in the middleof the axon as the current can only flow in one instead of two directions (or equivalently, thecurrent threshold is lower). Thus at the axon ends the variance in the voltage is double thatin the middle of the axon (Tuckwell and Walsh, 1983), making it the preferred initiation site.Because of the exponential dependence of spike rate on voltage variance, Eq.1, many morespikes will be produced at the axon ends than in the middle, despite the noise only beingdouble. Also note that the preferred initiation zone extends only little into the axon, muchless than the electrotonic length ⇤, which is about 80µm (for a 0.1µm diameter HH axonat rest, including the contributions of Na and K channels). The reason is that the channelnoise, which contains high frequencies, is filtered within a much shorter distance than theelectrotonic length ⇤, which describes the DC decay in a cable.

The total spontaneous generation rate in an axon of arbitrary length is by definition asum of the generation rate at the ends and the generation rate anywhere else. We define r

e

as the generation rate at one end (measured in Hz, taken within 40µm from the end) and⇢m

as the generation rate anywhere else (measured in Hz/mm), so that the rate at whichspikes are created, f

created

, can be written as

fcreated

= 2re

+ L⇢m

(2)

where we assume the axon is long enough (L � ⇤) so that we can ignore the length depen-dence of r

e

in the left portion of Fig.1A.These spike initiation rates, r

e

and ⇢m

, are compared as a function of diameter d inFig.1C. It shows that both decrease for wider diameters. The dependence on diameter can beunderstood as follows: the electrical coupling between neighbouring compartments increasesfor wider axons as the axial conductance is larger. Assuming a fixed channel density, wideraxons have more channels and hence more noise per length. As a result a larger diameteraxon collects more noise current, however, the impedance is lower as well, so that the netvoltage variance is smaller for wider axons. The effective area equals A / d⇤ / d3/2.

To test this we simulate a 1mm long axon and various diameter and extract the numberof spikes generated at the ends and elsewhere. The spike generation rates of the simulationsare well fitted to � log(r

e

) / (d/d0)3/2, while � log(⇢m

) / 2(d/d0)3/2, with a fit parameterd0 ⇡ 0.1µm. The factor 2 in the fit of ⇢

m

expresses the increased generation rate at theend of the axon explained above. Due to the exponential dependence of the rate on thefluctuations, the ratio of these generation rates is not constant, but shifts in the favor ofend-generation for thicker axons, thus contributing to the change in shape of the curves inFig.1A as the diameter is varied. We define the critical length L

c

as the length at whichthe two rates are comparable, L

c

= 2re

/⇢m

. It increases approximately exponentially withdiameter in our simulations, Fig. 2B. For very wide axons, which are better approximatedby a single compartment than a infinite long cable, we would expect � log ⇢

m

/ dL. Thishappens when d ⇡ 10µm for a 1mm long axon. Also for very narrow axons this theory willbreak down, because spike rates become so high that refractory effects become importantand the escape problem analogy no longer holds.

Collisions So far we have ignored another important effect for long axons, that also con-tributes to the shape of Fig. 1A: Two spikes traveling in opposite direction in the axon will

3

0 100 200 300 400 500

Distance along axon (µm)

0

2

4

6

8

Spik

es

genera

ted (

Hz)

10 100 1000 10000 100000

Axon length (µm)

0

10

20

30

40

50

Spo

nt r

ate

(Hz)

0.1 0.15 0.2 0.25

Diameter (µm)

0

5

10

15

20

Rat

e (H

z)

Initiation rate at endInitiation elsewhere (per mm)Fits

C D

A B

Figure 1: Spontaneous action potentials in Hodgkin-Huxley axons with uniform, fixed chan-nel densities as a function length and width. A) The total spontaneous action potential rateas function of axon length for diameters (top to bottom curve): 0.1, 0.11, 0.12, 0.13, 0.14,0.15, 0.2 and 0.25µm. B) The spike generation rate along an axon. Compartments at theends are hot spots that generate many more spikes per compartment than the rest of theaxon. Axon of 500µm length and 0.1µm diameter. C) The spontaneous spike initiation ratesat the ends (r

e

, in Hz) and in the rest of the axon (⇢m

, in Hz per mm) versus the diameter.Extracted from a simulation of an axon of 1mm length. Thin lines indicates fits to theory(see text). At the diameter at which the curves intersect, there are as many spikes generatedat one end as there are in the rest of this 1mm axon. D) Short segment of a simulationshowing the generation of two spontaneous spikes (arrows).

4

0.1 0.15 0.2 0.25

Diameter (µm)

103

104

105

106

Lcr

it (µ

m)

Lc (end vs rest of axon)Lcoll (collisions)

0 5000 10000

Axon length (µm)

0

25 Total observed

0

50

100

Ra

te (

Hz)

Generated elsewhereCollisions

0

25

50

Generated at ends

A B

Figure 2: A) Split of the various contributions to the observed spontaneous rate as a functionof axon length (axon with 0.1µm diameter). Top: rate of spikes generated at the axon ends.Middle: rate of spikes generated in the rest of the axon and the rate of collisions of spikes inthe axon. Bottom: The resulting observed firing rate (the sum of the two generation ratesminus the collision rate).B) The two characteristic length parameters as a function of axon diameter: Solid line (L

c

),for axons shorter than this length, most spikes are generated at the ends; for axons longerthan this length, most spikes are generated in the rest of the axon. Dashed line (L

coll

) foraxons longer than this length, collisions of spikes significantly limit the observed spontaneousrate.

collide. When this happens, the spikes annihilate as the membrane on either side of thecollision will be in a refractory state and further propagation fails. The probability thatspikes collide increases in longer axons: First, the rate increases with length as we have seenin Eq. 2. Secondly, it takes longer for the spikes to propagate the whole length. The speedof propagation in the HH model is about 200µm/ms for a diameter of 0.1µm, and increaseswith diameter as

pd. Thus while for short axons with low rates all initiated spikes travel the

full axon, in long axons the observed spontaneous rate is lower than the initiation rate dueto collisions. Note that in the absence of collisions, the observed rate is clearly independentof the observation point, whether at any end of the axon or anywhere in the middle, e.g.see Fig.3Ai. Yet it can be observed from diagrams like Fig.3Aii that also in the presence ofcollisions the observed rate of spikes will be the same at any point of the axon.

With every collision two spikes merge into one. The observed spontaneous rate, fobs

,thus equals the number of spikes generated at the ends, plus the spikes generated anywhereelse, minus the collision rate r

colls

,

fobs

= 2re

+ L⇢m

� rcolls

In Fig.2A we plot these terms as extracted from the simulations. For the shortest axons,most spikes are generated at the ends. With longer axons the rate of spikes generatedanywhere else increases linearly with the length, but the collision rate also increases linearly.

Collisions become significant if it is likely that there are multiple spikes propagatingsimultaneously in the axon. This happens when f⌧ ⇡ 1, where ⌧ is the time for a spiketo propagate from end to end ⌧ = L/v. With Eq.2 this gives the axon length for whichcollisions become important, L

coll

, as

Lcoll

=

1

⇢m

⇣�r

e

+

pr2e

+ ⇢m

v⌘

This is plotted in Fig.2B. For the HH system this length is always larger than the length for

5

which spikes generated in the rest of the axon are dominant. Thus when collisions start toappear, the spontaneous rate is dominated by the spikes generated in the rest of the axon,and so L

coll

⇡pv/⇢

m

. As explored in detail below, when L � Lcoll

the firing rate becomesindependent of the axon length, as can be seen in Fig.1A and Fig.2A.

In summary, there are 4 regimes in the HH axon that explain the shape of the graph ofspontaneous rate versus axon length, Fig.1A:

1. L ⌧ ⇤: the spontaneous spikes are mainly generated at the ends with a rate thatdecreases with length, no collisions occur,

2. ⇤ ⌧ L ⌧ Lc

: the spontaneous spikes are mainly generated at the ends, no collisionsoccur, the rate is roughly constant,

3. Lc

⌧ L ⌧ Lcoll

: the spontaneous spikes are mainly generated in the rest of the axon,no collisions occur, the spontaneous rate increases linearly with length,

4. L � Lcoll

: the spontaneous spikes are mainly generated in the rest of the axon, mostspikes collide, and the rate is independent of length.

This is an idealized description that is meant to aid understanding, in practice the regimeswill overlap. Changes in the ion channels properties and densities as well as the geometrycan further alter the classification and ordering of regimes. For instance, if the propagationspeed is slower, collisions will be more likely (L

coll

decreases). Or, as another example, ifone axon end has a soma attached to it, or it receives strong inhibition, no spontaneousspikes are expected to originate there. While in contrast, local depolarization can increaserates.

Effect of collisions in simplified models

In order to better understand the effect of collisions in long axons, we develop an abstractmodel of the action potential generation and propagation. Propagation failure is assumedto be negligible, indeed we never observed it in our Hodgkin-Huxley simulations. Spikegeneration probability is a function of the position along the axon. A simple model is tohave a rate r

e

at both ends, and a rate per length ⇢m

along the rest of the axon, i.e. thePoisson rate density is ⇢(x) = [�(x) + �(x � L)]r

e

+ ⇢m

. This system is simulated with ancellular automaton using a discretized axon, using 3 rules: It 1) creates spikes according toa Poisson process according to density ⇢(x); 2) it propagates the generated spikes towardsleft and right from the initiation point; 3) it annihilates colliding spikes.

Generators on axon ends only

We first analyze the idealized case when spikes are generated at the ends only. We assumethat the axon length is long enough so that r

e

is constant. As above we define the propagationtime ⌧ = L/v as the time it takes to travel from one end of the axon to the other. The onlydimensionless parameter is r

e

⌧ . We find (Appendix) that due to collisions the observed ratebecomes

fobs

= re

✓1 +

1

1 + 2re

⌧

◆(3)

For low rates, collisions should not occur, hence fobs

= 2re

. However, at high rates andlong ⌧ , every spike generated on one end will annihilate with one from the other end, hencefobs

= re

. The characteristic length for the switch-over is v/re

.

6

2δlδ l

a

iiiii

1

23

0.1 0.15 0.2 0.25

Diameter (µm)

0

20

40

60

Rate

(H

z)

Observed, HH 100mmPrediction, infinite length

Time

iA

B

Figure 3: A) Cartoon of: i) Unhindered propagation of a single spontaneous spike, ii)Illustration of two spikes colliding, the third spike does not collide, iii) Diagram used for thecalculation of the spontaneous rate in the limit of very long axons or high generation rates.B) Predicted spontaneous rate for axon of infinite length (dashed line, Eq. 6) and the rateextracted from HH simulations with 100mm length (solid). We extracted the propagationspeed v and the generation rate ⇢

m

from the simulation.

7

Uniform generation along the axon

Next, we assume constant and uniform generation along the axon (a so-called Poisson sheet).In the absence of collisions, the spontaneous rate obeys f

obs

= ⇢m

L, Eq.2. The first cor-rection will be the occasional collision of two spikes, see Fig. 3A.ii, the probability for suchevents is proportional to the rate squared. It can be calculated exactly (see Appendix)

fobs

= ⇢m

L

1� 1

3

⇢m

L2/v +O(⇢2m

)

�(4)

While in principle one can try to systematically expand to include higher order corrections,this quickly becomes unwieldy. Fig. 3A.ii illustrates why this calculation is hard: if thetop left spike (2) had not occurred, the top-right spike (3) would have collided with thebottom-left spike (1). In other words, the collision pattern can depend on the full spikehistory.

But while a general expression of spontaneous rate in terms of ⇢m

seems out of reach,the output rate becomes independent of the length of the axon in the limit of long axons(L � L

coll

). This can seen as follows: when the rates are high, most spikes will not propagatealong the full axon, instead, they will collide. Apart from the ends, the axon behavesuniformly. Because the probability for collision is approximately uniform and constant, thespike length l, i.e. the distance that a spike travels from initiation to collision point, will bedistributed exponentially

P (l) =1

�exp(�l/�) (5)

where � is the mean distance a spike travels before it collides.Next, we wish to know the dependence of this mean distance on the generation rate. It

follows from a self-consistency argument: Consider the spike labeled ’a’ in Fig.3A.iii. Thecollision process can be seen as a hazard-process, where the hazard for a collision is given bythe presence of other spikes. The hazard probability h is the probability for the presence ofanother spike in infinitesimal length �l. Consider the shaded strip in Fig.3A.iii. In the limitof small �l the colliding spikes present in this strip can be described by a binary randomprocess with an on-rate that equals the generation rate, 2⇢

m

�l, where the factor 2 is ageometrical factor, Fig.3A.iii. The off-rate of the process is the reciprocal of the mean timebetween collisions v/�. As required by self-consistency, this process yields colliding spikeswith a length that is exponentially distributed and mean length �.

The hazard is thus h =

2⇢m

�l

2⇢m

�l+v/�

⇡ 2⇢m

��l/v. This hazard will reduce the survivalprobability of the original spike according to P (l + �l) = P (l) � hP (l), so that P (l) /exp(�lh/�l). Identification with Eq. 5, gives the 2⇢

m

�/v = 1/�, from which the mean spikelength as a function of initiation rate follows as � =

pv/2⇢

m

.The final step is to compute the output rate from the mean spike length. Imagine that

one observes the spike rate at a given location in the axon. The average number of observedspikes will be product of the total number of initiations (⇢

m

L), the probability to havepropagated to that location (�/L), and finally a factor 2 because every initiation leads totwo spikes traveling in opposite direction. Thus, we have

fobs

= 2�⇢m

=

p2v⇢

m

(6)

This matches the simulations of the reduced model (not shown), but also the full HH modelwell, Fig.3B. Note that because L

coll

� Lc

for the HH axon, the uniform generation as-sumption is valid. Perfects fits should not be expected as for very narrow axons refractoryeffects will limit the rate in the HH model, while for very wide axons the length would needto be even longer than 100mm to ensure that we are in the collision dominated regime.

Variability of spontaneous spikes

In single compartments and axons much shorter than Lcoll

the spontaneous spikes in thereduced model will follow Poisson statistics and thus the inter-spike intervals have an ex-

8

0 0.2 0.4 0.6 0.8 1

Spike interval (s)

0

2000Opposite

0

2000

Count

Same direction

0

2000

Measured halfway

All spikes

0

2000

Count

Measured at end

Figure 4: Inter-spike time interval histograms in an axon with length 10Lcoll

. Top: countingall intervals; middle: events where the subsequent spikes travel in the same direction; bottom:events where the subsequent spikes travel in opposite direction. The variability is sub-Poisson (CV

isi

⇡ 0.67) in the middle of the axon (left column), but more Poisson-like atthe end of the axon (CV

isi

⇡ 0.78, right column). This is due to the fact that most spikesthere travel in the same direction (towards the end). Virtually no events have alternatingdirections at the end of the axon (bottom right panel).

9

ponential interval distribution. However, in longer axons the spontaneous spikes have moreregular statistics. The intuition is that when by chance at some time many spikes are gen-erated, the collision rate will increase as well, thus smoothing out some of the variabilityin the rate. To study this separately from the HH dynamics, including refractoriness, andmainly for reasons of efficiency we use the simplified Poisson sheet model, with a lengthL = 10L

coll

with a spontaneous rate of about 10Hz, assuming that spikes are generateduniformly throughout the axon (⇢(x) = ⇢

m

).Due to collisions the inter-spike interval distribution is more regular than a Poisson

process, Fig.4 (top left panel). This can be further analyzed by sorting the spikes accordingto their direction of travel. In particular, while the probability for a leftward-going spike tobe followed by another leftward-going spike approximately resembles a Poisson process, aleftward-going spike is unlikely to be immediately followed by a rightward traveling spike,as the second spike is likely to collide with a recent spike traveling in the opposite direction.As a result the probability for intervals where the spikes travel in opposite directions issmall for short intervals (bottom left panel, numerically resembling a gamma-distributionwith shape parameter k = 2). When measuring the interval statistics irrespective of thepropagation directions, the mix of these two processes leads to a process of sub-Poissonvariability with a interspike interval coefficient of variation (CV) of about 0.67. While awayfrom the ends, the rates of left-going and right-going spikes are equal, near the ends ofthe axon only few spontaneous spikes travel towards the middle. As a result the intervaldistribution is dominated by spikes traveling in the same direction and is more Poisson-likenear the ends and has a higher CV, Fig.4 (right panels). We also examined the effect of axonlength. The CV does not drop further for longer axons, demonstrating that the system iswell approximated by the collision dominated regime. For shorter axons the CV approachesone (more Poisson-like).

Effect on signal transmission

Finally, still using the Poisson-sheet model, we consider a situation where the neuron triesto send informative spikes down the axon. We analyze the spikes arriving at the other endof the axon where one could image a synaptic terminal to be present. We consider thetransmission of a short burst of spikes through the axon and measure whether or not thespikes arrive at the terminal at the expected time, Fig.5A. As one can observe the first spikehas only a limited probability to be transmitted, while later spikes are more likely to betransmitted, Fig.5B. The reason is that the first spike has a high probability to encounter aspontaneously generated spike and thus be annihilated. The first spikes act as a snowplow,clearing the way for the subsequent spikes.

The probability that a colliding spike is present is proportional to the area of the stripedtriangle in Fig.4C, which equals ⇢

m

L⌧ = (L/Lcoll

)

2. Under a Poisson assumption, theprobability for transmission without collision equals exp[�(L/L

coll

)

2]. For a spike much

later in the train the collision probability is equal to the slice P = ⇢m

Ltisi

, where tisi

is thetime between the spikes in the train, which is much smaller (hatched region in Fig.4B). Thetransmission probabilities for Fig.4B are accordingly 0.37 for the first spike and 0.8 for thelater spikes (for the 50Hz train), which matches the simulations reasonably well. The higherthe frequency of the train, the less probable it is that a spontaneous spike occurs betweenthe spikes, hence the reliability increases with the frequency of the train, Fig.4B. This isdifferent from most other types of noise in spiking, as usually the first spike is the mostreliable, while subsequent ones are either jittered or unreliable (e.g. Mainen and Sejnowski,1995; van Rossum, O’Brien, and Smith, 2003).

In addition to the spike times of the arriving train one can study the total spike count. Wemeasure the output rate as a function of the driving rate for a long train. The spontaneousactivity will cause output spikes in the absence of somatically driven spikes, abscissa inFig.5D. However, unlike a single compartment model, due to collisions the output rate isnot simply the sum of driven and spontaneous spikes. Instead, the driven spikes partly

10

Time0 0.5 1 1.5

Time (s)

Driven end

Output end

1 5 9

Index of spike in the train

0

0.25

0.5

0.75

1

Tra

nsm

issi

on

re

liab

ility

f=100f=50f=25

0 10 20 30 40

Stimulus rate (Hz)

0

10

20

30

40

50

Outp

ut ra

te (

Hz)

Long axon Very short axonMaximum model

A

DB

C

Figure 5: A) Raster plot of a short simulation where the spike trains are generated at oneend of the axon (top) and received at the other end of the axon (bottom). The short dashedbars in the lower row indicate when spikes would arrive had they not collided. In particularthe first spikes in the train are vulnerable to transmission failure. Some spontaneous spikescan be observed at either end.B) The transmission probability of a spike from one end of the axon to the other for variousstimulus frequencies. Due to collisions the first spike has a low transmission probability, butlater ones encounter only few spontaneous spikes on their way and have a high transmissionprobability, in particular at high stimulus frequencies.C) Diagram to calculate the transmission probabilities. The first (left-most) spike in thetrain has a high collision probability (shaded triangle region), but later spikes have a lowerone (hashed region).D) The output rate as a function of stimulus rate in a long axon (thick solid curve). Lowstimulus frequencies hardly increase the output rate. In contrast, in a single compartmentmodel with spontaneous Poisson spikes, the output rate is simply the sum of spontaneousand driven rates (dashed line). Axon of length L = L

coll

, 10Hz spontaneous rate.

11

collide with the spontaneous spikes. As a result of this ’replacement noise’, the resultingrate resembles the smoothed maximum of spontaneous and driven rates, Fig.4D (thick solidcurve). For comparison the mathematical maximum of spontaneous and driven rates isshown by the thin line. Due to the replacement noise small stimuli will hardly increase theoutput rate and the Signal-to-Noise ratio for such stimuli will be particularly poor.

Discussion

Some effects of ion channel noise have previously been studied in dendrites and axons. Forinstance, as the spike propagates, the speed varies, resulting in jitter of the spike arrival timesat the axon terminals (Verveen and Derksen, 1968; Lass and Abeles, 1975; Horikawa, 1991;Faisal and Laughlin, 2007). We have considered spontaneous spike generation in axons. Inthin axons the spontaneous rates can be substantial. The spontaneous spikes are preferablygenerated at the ends of the axons, with a much lower rate in the rest of the axon. However,for long axons (L � L

c

) the contribution of the rest of the axon dominates. As a resultwe find a non-monotonic relation between the spontaneous rate and the length of the axon.The minimal firing rate occurs at a length of about 1mm, which accidentally coincides withthe length used in Faisal, White, and Laughlin (2005). Therefore our results put even morestringent bounds on the minimal diameter to prevent high spontaneous rates than suggestedthere. In particular long axons might need to be somewhat thicker.

For even longer axons (L � Lcoll

), we find that the spontaneous rate saturates as spikeswill start to collide with each other. Our theory predicts when this will happen and whatthe expected spike rate will be. Many questions are left about the collision regime, even inthe simplified model: What is the firing rate for intermediate length axon, between Eq.4and Eq.6? What if refractory time is included? And what is the exact interval distributionfor such axons?

The most prominent question is whether spontaneous spikes caused by stochastic pro-cesses occur in biology and at what rate. If there is strong evolutionary pressure to reducewiring diameter, as has been suggested (Faisal, White, and Laughlin, 2005), it is not unlikelythat the nervous system operates in a regime where some spontaneous spikes occur.

Related to that question is our choice of model system, namely the standard Hodgkin-Huxley model. The simulation presented in Faisal, White, and Laughlin (2005) show thatthe spontaneous rates in HH and axon models tuned to mammalian fibers are comparable.Nevertheless, the exact characterization of the voltage-gated channels and their densities inlong axons required to make quantitative predictions is far from complete and the sponta-neous rate’s exponential dependence on the fluctuations, Eq.1, makes it sensitive to precisemodel details. Whether spontaneous spikes are mainly generated throughout the axon orat the ends is similarly difficult to determine. In this light, the current study should beseen as demarcation of the various possible regimes. However, our results on the intervalstatistics and transmission might be used to test whether axons are in the collision regime.The conduction speed in mammalian C-fibers is on the order of 1 m/s (cf. to 0.2m/s for theHH model), so that for spontaneous rates of 1Hz, the collision length is L

coll

⇡1m.

Acknowledgments

Discussions with Sue Fleetwood-Walker, Jean-Marc Luck and Eleni Vasilaki are gratefullyacknowledged. COD was supported by EPSRC/BBSRC/MRC through the Doctoral Train-ing Centre in Neuroinformatics.

12

L

0

Time

2τ

L

0

Figure 6: Top: When spikes are generated at the ends only, a second spike generated at thebottom on the indicated interval 2⌧ will collide with the top spike. Right: note, that if thefirst top spike has collided, a second top spike propagates freely.Bottom: When spikes are generated throughout the axon, a second spike will collide withthe indicated one if it falls anywhere in the shaded triangle region.

Appendix: Analytic approach

Spontaneous rate for generation at ends only

To calculate the spontaneous rate when spikes are generated at the ends of the axon only,we use the following approach. We assume that during a time T there is total of n spikesgenerated of which m are on one end of the axon and n�m at the other side. Both m andn � m are independently Poisson distributed. In this case each spike can at most collidewith one other spike. We calculate the probability that a given spike at the top end collideswith any of the n�m spikes from the other end. The probability for a given top-end spiketo collide with a bottom-end spike is 2⌧/T , Fig.6top. Because the events are independentthe probability for colliding with any spike is p0 = 1� (1� 2⌧/T )n�m. We ignore effects atthe start and end of the time, which is allowed as we will take the limit T ! 1.

When a collision happened, the collided spikes should be removed from the pool andthe subsequent collision probability changes. For instance, if m = 2, there are two possibleways that just one collision takes place: 1) The first spike does not collide (probabilityq0 ⌘ 1 � p0), and the second does (probability p0), and 2) the first spike collides p0, andthe second does not. However in the second case we have to update the collision probabilityas there is one spike less to collide with. The probability for the second spike not collidingbecomes q1 = 1� p1, where p

i

= 1� (1� 2⌧/T )n�m�i ⇡ 1� exp[�2(n�m� i)⌧/T ] is theprobability for collision after i spikes have collided already and have thus been removed.

The probability for k collisions in time T given m spikes originating from one end and nfrom the other end is

p(k|m,n) =

k�1Y

l=0

pl

!m�kX

i

o

,i1...ik=0Pl

i

l

=m�k

"kY

l=0

qill

#(7)

The term in the round brackets corresponds to the probability of k spikes colliding, whilethe sum term corresponds to all configurations of other spikes not colliding. The sum runsfrom 0 to m� k for each index i, but is restricted so that the sum of indices

Pl

il

= m� k.This ensures that there are exactly m� k spikes that do not collide. The average observedrate follows as f

obs

= (n� hki)/T .Eq. 7 is a polynomial in ⌧ , that quickly grows in complexity with increasing k and m.

Despite having an explicit expression for p(k), a systematic expansion for hki in the limit of

13

large n,m, T is hard. Using Mathematica we numerically expanded the mean k wrt to ⌧/Tand find that the terms behave as hki = m[2m⌧/T � 4(m⌧/T )2 + . . .], so that the numberof observed spikes is n� hki ⇡ 2m� 2m(m⌧/T ) + 4m(m⌧/T )2 + . . . matching the relationfound above, Eq.3.

Perturbation expansion for spontaneous rate for uniform generation

Here we derive Eq.4, the collision rate in the first order in the limit of low rates assuming aconstant, uniform generation throughout the axon with a rate ⇢

m

, although the formalismcan in principle be extended to varying rates. The mean number of spikes generated duringa time T is n = ⇢

m

LT . So that the observed rate will be ⇢m

L if there are no collisions.The first correction corresponds to two spikes colliding so that only one spike remains.

Consider a first spike generated somewhere, sometime in the axon, Fig. 6bottom. If thesecond spike is generated in either shaded triangular region, it will always collide with thefirst spike. Given a spike generated at a time t1 and location x1, the probability for acollision with the second spike (x2, t2) is given by the integral over these shaded regionsdivided by the total area, LT . This can be formally expressed using integrals with theHeaviside function H() as

p(x1, t1) =

1

LT

ˆx1

0dx2

ˆT

0dt2H(x1/v + t1 � x2/v � t2)H(x1/v � t1 � x2/v + t2) +

ˆL

x1

dx2

ˆT

0dt2H(�x1/v � t1 + x2/v + t2)H(�x1/v + t1 + x2/v � t2)

Averaged over all possible positions of the first spike and in the limit of T ! 1,

p =

1

LT

ˆL

0dx1

ˆT

0dt1p(x1, t1)

=

2

3

⌧

T

This is the collision probability per spike pair. As there are n2/2 pairs, the collision rate isfcoll

= n2⌧/3 = ⇢2m

L3T/3v, which leads directly to Eq.4.The same arguments can be followed for the possible collisions with three spikes. Ob-

viously, two spikes can collide, while a third one propagates independently (order ⇢2m

).However, the interaction of three spikes (order ⇢3

m

) can lead to either one or two collisions,depending on the precise configuration of the initiation points. These effects make higherorder expansion complicated and cumbersome.

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