Reliable circuits from irregular neurons: A dynamical approach to ...

J. Physiol. (Paris) 94 (2000) 357–374© 2000 Elsevier Science Ltd. Published by Editions scientifiques et medicales Elsevier SAS. All rights reservedPII: S0928-4257(00)01101-3/REV

Reliable circuits from irregular neurons: A dynamical approachto understanding central pattern generators

Allen I. Selverston*, Mikhail I. Rabinovich, Henry D.I. Abarbanel, Robert Elson,Attila Szucs, Reynaldo D. Pinto, Ramon Huerta, Pablo Varona

Institute for Nonlinear Science, Uni6ersity of California, San Diego, 9500 Gilman Dri6e, La Jolla, CA 92093-0402, USA

Abstract – Central pattern generating neurons from the lobster stomatogastric ganglion were analyzed using new nonlinear methods.The LP neuron was found to have only four or five degrees of freedom in the isolated condition and displayed chaotic behavior. Weshow that this chaotic behavior could be regularized by periodic pulses of negative current injected into the neuron or by couplingit to another neuron via inhibitory connections. We used both a modified Hindmarsh-Rose model to simulate the neurons behaviorphenomenologically and a more realistic conductance-based model so that the modeling could be linked to the experimentalobservations. Both models were able to capture the dynamics of the neuron behavior better than previous models. We used theHindmarsh-Rose model as the basis for building electronic neurons which could then be integrated into the biological circuitry. Suchneurons were able to rescue patterns which had been disabled by removing key biological neurons from the circuit. © 2000 ElsevierScience Ltd. Published by Editions scientifiques et medicales Elsevier SAS

central pattern generator / neural modeling / nonlinear systems electronic neurons / chaotic behavior

1. Introduction

Central pattern generating circuits (CPGs) areneural networks specialized for the production ofrhythmic motor patterns. Found in all nervoussystems, they have evolved to produce rhythmicspatio-temporal motor patterns independently, i.e.without the need of rhythmic sensory feedback [4].Of course the patterns are strongly influenced bysensory feedback as well as by chemical neuro-modulatory substances, but the fundamental struc-ture of the pattern is determined by the synapticconnections and biophysical properties of the indi-vidual neurons. These properties are crucial indetermining the timing and organization of themotor patterns. They can be altered chemically sothat, by exposing CPGs to chemical modulatorsalways present in nervous systems, they can befunctionally reconfigured, so that one anatomicalcircuit can produce many different stable modes ofactivity. This presents an analytical challenge sincein intact animals the properties of CNS neuronsare in a constant state of flux [9]. In general, CPGsrequire the presence of one or more modulators tofunction in a normal manner and if completelyfree of modulatory influence will usually be inoper-ative. So the challenge is to provide CPGs with

enough modulation to be rhythmically active andthen hold the chemical environment constant sothat the rhythms will be stable for extended peri-ods. Depending on the particular chemical modu-lators present, a large number of different patternscan be generated. During these stable regimes,however they must still be flexible enough to re-spond to sensory input from the environment.Most circuits of model neurons are unable tosimulate this important property.

Invertebrate CPGs have been shown to be veryuseful for studying the mechanisms of patterngeneration because they are particularly amenableto experimental analysis and they are usually madeup of limited numbers of relatively large neuronsthat can be repeatedly identified from one animalto another. The part of the nervous system con-taining the CPG machinery can be removed fromthe rest of the animal, thus reducing or entirelyeliminating the changes caused by the chemicalsubstances they are exposed to normally. Usingsmall CPG systems, paired recordings can be madefrom the same identified neurons in differentpreparations and their synaptic connectivity estab-lished. Recording from pre- and postsynaptic neu-rons has uncovered the circuitry for many smallCPGs. We suggest that the neural patterns theseCPGs generate are robust and reliable in the faceof noise and external perturbations even though,as we will demonstrate, individual neurons in thecircuit may behave chaotically. By behaving chaot-

* Correspondence and reprints.E-mail address: [email protected]

(A.I. Selverston).

A.I. Selverston et al. / Journal of Physiology 94 (2000) 357–374358

ically we mean that when isolated from all synap-tic input from other neurons and subjected to bothlinear and nonlinear analysis, they not onlydemonstrate irregularities in their membrane po-tential bursting/spiking firing patterns, but alsodemonstrate deterministic chaos in their phasespace attractors, Lyanopov exponents and entropymeasurements.

To investigate this thesis, we have begun toanalyze the pattern generating mechanisms of thelobster pyloric rhythm using new analytical tech-niques. We have been working on two CPGs, thegastric and the pyloric, from the lobster stomato-gastric system, for over 30 years [9, 18, 19]. Thegastric CPG is the more complex of the two andoperates the three teeth of the gastric mill. Thepyloric CPG is the faster rhythm and operates afiltering mechanism at the lower end of the stom-ach. Although driving movements of the stomach,the muscles involved are striated and require con-tinuous control from the CNS. While the physio-logical details of both circuits are known, it cannotbe said that we understand, by any rigorous for-mulation, how the motor patterns are produced,i.e. at a reductionistic level we know the physiolog-ical properties of the cells and synapses, but thetotal operation of the network is an ‘emergentproperty’ that is more than the sum of its parts.This is especially the case for CPGs where most ofthe properties are nonlinear and dynamic.

We have analyzed the pyloric CPG because webelieve that by understanding the patterns pro-duced by this small system, we will gain someinsights as to how larger circuits in the brain areable to generate spatio-temporal patterns. Our ap-proach relies heavily on being able to move seam-lessly between experiment and theory. At thepresent time it is not possible to prove or disprovethe validity of large scale modeling of the brainsince untestable assumptions are made with re-gards to synaptic connectivity and cellular proper-ties, general phenomena which are essential inmaking useful and informative models. We seek toprovide a rigorous formulation of the underlyingprinciples by modeling a well-defined system. Ourmodels mimic the output of a biological CPG notjust in the steady state, but during a variety ofperturbations as well. We are particularly inter-ested in capturing the dynamical properties of thebiological systems. The models are simple enoughto study with available mathematical and compu-tational techniques. We have been applying thesenew modeling and nonlinear analysis methods tothe lobster pyloric CPG in order to try and answerseveral fundamental questions:

– How can a CPG, made up of neurons whichproduce chaotic voltage oscillations, generate regu-lar spatio-temporal rhythmic activity patterns thatare stable, robust and reliable in the presence ofnoise, yet flexible enough to provide specific andreproducible responses to inputs?– If there is chaos in CPG systems, what role doesit play during normal behavior?– How many degrees of freedom are needed formodeling a biological CPG accurately? i.e. howmany equations will capture the biologicalbehavior?– We know that in intact systems neurons firemore regularly than when isolated. What are themechanisms that achieve this regularization in in-tact GPG systems?

Our strategy has been to first determine thedynamical properties of single neurons in the py-loric circuit after they have been synaptically iso-lated from the other neurons. Thus, if the isolatedneurons show chaotic behavior, we can be sure itis not the result of nonlinear synaptic interactions.We then examine how these isolated neurons canbecome regularized by various kinds of synapticinputs from other neurons or by periodic pulses ofcurrent. We then extend the analysis by construct-ing mathematical models of the pyloric neuronsthat reproduce the dynamical aspects of isolatedcells and small subsets of cells both in steady stateand perturbed conditions. Finally we describe anin silico model of neurons, i.e. electronic neuronsthat though elegantly simple, appear to capture allof the essential dynamical features of membranepotential in their biological counterparts. We ex-amine the properties of their individual behaviorsunder a variety of perturbations as well as examin-ing their behaviors in pairs and in hybrid circuitswith biological neurons.

2. Dynamical properties of biological neurons

Neurons in the pyloric CPG circuit are synapti-cally connected as shown in figure 1. There arefourteen neurons in the circuit, all motor neuronsexcept AB, which is an interneuron. When thecircuit receives neuromodulatory input from twoupstream ganglia, the commissurals (CGs), themembrane potentials of all of the neurons showbursting-spiking activity, i.e. they are what aretermed conditional oscillators, conditional uponthe presence of modulators that activate the under-lying bursting currents. The oscillatory activity issomewhat irregular in each individual neuron, but

A.I. Selverston et al. / Journal of Physiology 94 (2000) 357–374 359

Figure 1. Electronic equivalent circuit for the lobster pyloriccentral pattern generator. Black dots represent chemicalsynapses and resistors represent electrotonic connections. Thereare two pyloric dilator neurons (PD), one lateral pyloric (LP),one ventricular dilator (VD), one inferior cardiac (IC) and eightpyloric neurons (PY). All of the neurons except the anteriorburster (AB), are motor neurons as well as pattern formingneurons.

cuit. The AB interneuron is the only neuronwhich, when isolated, has periodic regular burstingactivity. Because the AB’s activity has the highestnatural frequency, it also serves as the pacemakerfor the entire pyloric network. The neurons com-prising the pyloric network are typical of bothvertebrate motorneurons and spinal interneuronsin that they contain large numbers of channels,receptors and synaptic mechanisms [9].

We selected the LP neuron for detailed analysisbecause it appeared to be typical of many pyloricmotor neurons and a large amount of informationabout this particular neuron already existed [8].Intracellular time series lasting several hours weresampled at a data rate of 2 kHz (figure 2:1). TheFourier power spectrum for this time series wasbroadband without separate peaks as is typical incases of a chaotic time series, and did not revealanything about the dynamics underlying the intra-cellular voltage signal. A nonlinear analysis how-ever suggested that by using the method of falsenearest neighbors [1], the active degrees of freedomfor the LP neuron are only four or five. Thissuggests we can make models of the behavior ofthis neuron with only five degrees of freedom. Thiswas a remarkable finding because neurons contain

becomes more regularized when the neurons areembedded in a network and receive synaptic input(mostly inhibitory) from other neurons in the cir-

Figure 2. Irregular spontaneous bursting in a synaptically isolated LP neuron still receiving modulatory input from the commissuralganglia. (1) Excerpt from an extended time series showing a typical irregular bursting pattern. (2) Three bursts aligned by their firstspike and superimposed. (3) Phase space diagram of slow voltage oscillations from a timeseries of a similarly isolated LP. Themembrane potential [V(t)] was low-pass filtered to remove the spikes. The plot of V(t) against its first and second derivativesreconstructs an attractor showing, in addition to noise, apparent deterministic structure. The motion in time is counter-clockwise. Thelarge orbit corresponds to the overall slow wave (from burst to burst), the smaller spiral to faster oscillations on the plateau (duringeach burst).


many more than five fast and slow dynamicalprocesses and therefore usually considerably morethan four or five equations are thought to benecessary for an adequate model. Modifications ofthe original Hodgkin-Huxley equations [12] to in-clude the dynamics of more recently discoveredchannels are a good example. This finding openedup two possible options for us to pursue. On theone hand we could use rather simple phenomeno-logical models that might capture the dynamics ofthe neuron but would be inappropriate for model-ing experimental data. On the other hand we couldutilize models which incorporate mathematical de-scriptions of the principle ionic channels necessaryfor more realistic simulations. We have tried bothapproaches.

When the analysis of the LP time series data wascomplete, it showed that the isolated neuron had apositive global Lyapunov exponent, a hallmark ofchaotic behavior in a dynamical system. In an-other study in which we made slight perturbationsto the orbit, we found that there were two positiveglobal Lyapunov exponents, one zero exponentsuggesting the behavior is governed by ordinarydifferential equations, and two negative exponents.A fractal dimension of approximately 4.4 was es-tablished based on these global exponents. Chaoticneurons in invertebrates have been reported previ-ously [10, 13] but their analysis was much lessdetailed. Chaotic oscillations are a possible state ofactivity when many ion channels having differentkinetic properties are operating, but in fact manytypical Hodgkin-Huxley type models show littlechaos.

One may speculate as to the actual usefulness ofchaos in small systems — the ability to explorewider domains of high dimensional space than ifconstrained to a limit cycle for example, but thefact is that when neurons are embedded in acomplex synaptic circuit, they behave in a muchmore regular way than when they are isolated. Thenetworks made up of these unstable neurons oper-ate in a reliable and predictible fashion but arerobust enough to resist strong perturbations.CPGs that produce rhythmic motor patterns forcyclic behaviors must contain a reliable system forproviding impulses to the appropriate muscles. Itis clearly important then to determine the mecha-nisms involved in achieving this degree of regular-ity and stability from what are inherently unstableelements without losing the ability of the CPGs torespond to sensory and command elements ortheir ability to restructure themselves into func-tionally new networks under the influence of neu-

romodulators. Therefore we decided to initiallyexamine the regularizing effects of various formsof current injection, both natural and artificial,into isolated pyloric neurons which were display-ing chaotic activity.

2.1. Regularization of chaotic neurons

We have studied the LP neuron by characteriz-ing its dynamic response to periodic current pulses,sinusoidal driving and phasic inhibitory inputfrom presynaptic pacemaker neurons. Experimen-tal perturbations were applied to isolated neuronsand the resulting time series were analyzed usingan entropy measure obtained from the power spec-tra. As previously noted, when the LP was isolatedfrom phasic inhibitory input, it generated irregularspiking-bursting activity. Each burst begins in arelatively stereotyped manner, but then evolveswith exponentially increasing variability (figure2:2) [6] This can also be seen in the chaoticattractor (figure 2:3).

2.2. Regularizion of chaotic acti6ityby current pulses

We found that depolarizing current pulses werepoor regulators of the neurons but the hyperpolar-izing pulses were able to exert a strong, frequency-dependent regularizing action. The LP wassynaptically isolated from other neurons in thepyloric circuit by photoinactivating the PD andVD neurons (because they are cholinergic andcannot be specifically blocked pharmacologically)and blocking the remaining glutamatergic synapseswith picrotoxin. The CGs were left connected tothe stomatogastric ganglion via the stomatogastricnerve so the isolated LP would receive the sameneuromodulators it receives under normal experi-mental conditions. Long time series traces (5–6min) of membrane voltage were digitized and thedegree of irregularity in the observed oscillationswas determined from the power spectra and ameasure of the entropy for the harmonics of thepower spectra for frequencies up to 40 Hz.

Fast Fourier spectra were calculated from thedigitized voltage signals without filtering and thepower in harmonics up to 40 Hz, P(n) for n=20 976 were converted to probability values asp(n)=P(n)/SP(n).

An entropy function (in bits), S=−Snp(n)log2[p(n)], was then evaluated. S gives aquantitative measure of the complexity of thepower spectrum associated with the observed time


Figure 3. Regularization of irregular bursting activity by injec-tion of periodic current pulses. Recordings (a–c) and analysis(d–f) of voltage activity in an LP neuron after removal ofchemical synaptic input from other pyloric neurons. (a) Freerunning control activity. (b) Forcing by periodic depolarizingcurrent pulses (frequency, 0.45 Hz). (c) Forcing by periodichyperpolarizing current pulses at the same frequency. (d)Fourier power spectrum for control activity. (e) Spectrum fordepolarizing pulses. (f) Spectrum for hyperpolarizing pulses.Entropy values (S) are given in bits. Conditions: VD and bothPDs killed; PTX; 1 mM 4-AP.

0.45 Hz, to approximate excitatory input, therewere changes to the timing and duration of thebursts as might be expected, but little change inthe irregularity (figure 3b). The entropy droppedonly to 7.71 and there was little change to thepower spectrum aside from the addition of localpeaks at the stimulus frequency and it’s multiples(figure 3e). Hyperpolarizing current pulses meantto mimic inhibitory input had a much more pow-erful effect on regularizing the bursts (figure 3c).As shown in figure 3c, f, entropy was reduced to3.46 bits and the power spectrum was concentratedinto sharp peaks at the applied frequency and itsharmonics.

The mechanism underlying the stabilization ap-pears to be due to the hyperpolarizing currentterminating the plateau potentials before instabili-ties in the burst phase occur. The initial stablephase of the burst can also be extended until thearrival of the next inhibitory input.

2.3. Regularization of LPby biological pacemaker inhibition

We also examined the way in which the LPneuron could be regularized by biological in-hibitory input. To do this we experimentally pro-duced a subcircuit consisting of the pacemakergroup consisting of the electrotonically-coupledAB and PD neurons, and the LP neuron. The VDneuron was photoinactivated and removed fromthe network and all other synapses once againblocked with picrotoxin. This left the PDs withtheir inhibitory inputs to the LP intact and bydirect current injection we could alter the fre-quency of bursting in the AB-PD pacemakergroup. If we examine the effects of the PD forcingat different frequencies (figure 4), it is clear thatthe spontaneous pacemaker input regularized thefree-running chaotic activity of LP in a frequency-dependent manner.

During spontaneous activity of the subcircuit,the LP displayed a power spectrum with strongpeaks at the PD spontaneous frequency and itsharmonics. The LP entropy was reduced to S=5.1bits [6]. The LP was also able to follow at lowerfrequencies but as the forcing frequency was re-duced, the LP power spectra broadened and loststructure. When the PDs were shut of entirely, theLP activity displayed a broadband structure andthe entropy increased to S=7.9 bits, essentiallythe same as above.

series. By this measure, a power spectrum with asingle peak (a linear periodic system) yields S=0bits. A spectrum with two isolated peaks (a linearquasi-periodic system with two possible states)S=1 bit, a flat or white noise spectrum extendingto 40 Hz (or n=20 976) yields S= log2(20 976)=14.35 bits. Low values of S, relative to 14.35 bits,are then interpreted as associated with lower com-plexity in the oscillations of the system.

As shown in figure 3a, d, an isolated but stillmodulated LP neuron behaves irregularly with anentropy S value of 8.08 bits and a power spectrumwhich shows a wide distribution and a broad peak.When current pulses were applied at a slightlyhigher frequency than the free-running neuron,


Figure 5. Reciprocal synaptic inhibition regularizes the chaoticbursting of LP and a single PD neuron. (A) Intrinsic bursting ofLP and PD neurons recorded (at different times in the sameexperiment) while isolated from circuit interactions. (B) Thesame two neurons coupled by reciprocal inhibition. The LP-to-PD connection is inserted via a dynamic clamp. Conditions: AB,VD and one PD photoinactivated; PTX. In (A), the LP record-ing was made with the remaining PD deeply hyperpolarized.

Figure 6. Regimes of oscillations in two coupled neurons. (a)With no applied current, the slow oscillations are synchronizedwhile the spikes are not (b) counteracting the natural couplingleads to independent bursting (c) with net negative coupling, theneurons burst in anti-phase in a regular pattern (d) whendepolarized by a positive depolarizing current, both neurons firea continuous pattern of synchronized spikes. In the figure, ga isthe added synaptic conductance and I the current injected intothe two neurons.

2.4. Mutual regularization by reciprocal inhibition

Can chaotically bursting neurons regularize eachother? When the AB neuron is killed, the PDs alsoburst irregularly (figure 5A). When PD and LPinhibited each other via their normal connectivity,they adopted anti-phase coordination and regu-larized their bursting (figure 5B). What with allnatural synapses blocked, and with simulatedsynapses inserted via the dynamic clamp [21], wecould show that only reciprocal inhibition (not anyother pattern of connectivity) could produce regu-larization (unpublished data).

2.5. Synchronization and regularizationby electrotonic coupling

Electrotonic coupling exists between many neu-rons in the STG. We studied the dynamical effectsof adding artificial electrical coupling in parallel tothe natural coupling between the two PD neurons,

again using a dynamic current clamp device. Theeffective coupling could be altered by injectingequal and opposite current (Ia) into each cell viamicroelectrodes such that Ia

( j )=ga(Vj−Vi), wherega is the added synaptic conductance and VI is themembrane potential at the soma PDI [20]. Typicalrecords for added conductances are shown in(figure 6) [5]. The natural coupling of these neu-rons synchronized bursts but not individual spikes(figure 6a). The natural coupling was sufficient tosynchronize spikes during the tonic firing (figure6d) implying a shunting action of slow membraneconductances during burst generation. Opposingthe natural coupling via the dynamic clamp causesthe bursting to become more asynchronous (figure

Figure 4. Control of irregular bursting by rhythmic synaptic inhibition: intracellular recordings and Fourier power spectra. In thisconfiguration, the only strong phasic synaptic input to LP comes from the PD neurons which are part of the pacemaker group. Therhythmic bursting of the pacemakers was altered by current injection. Left column: simultaneous intracellular recordings from LP andPD neurons. Right column: corresponding Fourier power [P( f )] spectra for the LP membrane potential, evaluated from long timeseries. The displayed frequency range encompasses the slower voltage oscillations. (A) Spontaneous activity of subcircuit. (B) PDforced to burst at 1 Hz. (C) PD forcing at 0.65 Hz. (D) PD forcing at 0.4 Hz. (E) Free running activity of LP when PD bursting wasshut off by strong hyperpolarization.


6b) finally leading to out-of-phase bursting andincreased regularization (figure 6c). In this case thenet negative coupling approximates reciprocalinhibition.

3. Computational and electronic modelingof STG neurons

Because our analysis of biological neurons hadshown they could be modeled with only three orfour degrees of freedom, we chose initially to use afamiliar simplified model put forward by Hind-marsh and Rose (HR) [11]. The general form ofthis model contains three terms:

dx/dt=ay(t)+bx2(t)−cx3(t)−dz(t)+I (1)

dy/dt=e− fx2(t)− f(t) (2)

dz/dt=m [−z(t)+S(x(t)+h)] (3)

The first three equations represent the originalmodel where x(t) corresponds to membranevoltage, y(t) represents a ‘fast’ current and bymaking m�1, z(t) a ‘slow’ current. The first threeequations (the 3-D model) can produce severalmodes of spiking-bursting activity including aregime of chaos that appears similar to that seen inbiological neurons.

However the parameter space for the chaoticbehavior is much more restricted than we observein real neurons. The chaotic regime is greatlyexpanded by incorporation of the fourth term intothe model:

dw/dt=n [−kw(t)+r(y(t)+1)] (4)

Adding this term w(t) to introduce an evenslower process (nBm�1) is intended to representintracellular Ca dynamics. Note this also adds a−gw(t) to the second equation. When this term istaken into account, the model produces simula-tions of intracellular activity that are even moresimilar to the biological observations. We, as yet,do not know if the w(t) term actually representsCa kinetics in pyloric neurons and experiments arecurrently under way to measure Ca transients inneurons using optical methods.

Because of its relative simplicity, the HR modelwas extremely useful in constructing an analogimplementation — an electronic neuron — thatcould perform the computations necessary to emu-late stomatogastric neurons in real time. Howeverthe down side of such a simplified model is that itis difficult to compare with biological neuronswhich are made up of a multitude of individual

conductances and compartments. Since the experi-mental manipulation of these parameters is crucialin establishing physiological mechanisms, we alsodeveloped a Hodgkin-Huxley (H-H) type modelthat is more biologically realistic than that ofHindmarsh and Rose. In this second model [7], theneuron is represented by two compartments, onefor the lumped neuropilar processes and soma andthe other for the axon. The currents we have usedare based on previous descriptions [3, 23] but arerestricted to those that we consider to be the mostimportant in generating slow wave and spikingcurrent. In all, five currents are present in thesoma-neuropil compartment (ICa1 low voltage acti-vated calcium current, ICa2 high voltage activatedcalcium current, Ih hyperpolarization-activated in-ward current, IK(Ca) calcium dependent potassiumcurrent, and a leak current IL). The axonal com-partment contains the well-known Hodgkin-Hux-ley currents for spiking – INa, IK, and a leakcurrent IL. The soma-neuropil compartment alsoincorporates intracellular dynamics based on themodel of Li et al. [13]. In this model, cytosolic(Ca2+) is determined by influx across the plasmamembrane and by uptake and release from the ERand extrusion by a plasma membrane pump and aplasma membrane Na-Ca exchanger.

3.1. Modeling of single neurons

The three-dimensional H-R model generates atime series that looks remarkably similar to thetime series of an isolated LP neuron (figures 2–4,7), and its strange attractor has the same topologyas the one shown in figure 2. Even without anequation for the slow Ca dynamics, the modelcontains the appropriate mix of slow and fastdynamics to accurately describe the bursting spik-ing behavior seen in single isolated pyloricneurons.

The soma voltage time series trajectories shownby the Hodgkin-Huxley-calcium model were alsoquite similar to those observed in biological neu-rons [8]. In both cases, the most common featurewas the variability in burst duration. In isolatedLP neurons, the burst periods were in the range of1–3 s and for the model in the range of 1.5–3 swith the most frequent period being about 1.7 s. Inthe H-H calcium dynamics model, the soma com-partment produces plateau depolarizations thatproduce spiking in the axon compartment. Caplays a key role in determining the length of theplateau and therefore the length of the burst. Theplateau potential is maintained in the model by thecompetition between the inward currents [7].


Figure 7. A Hodgkin-Huxley plus calcium conductance-type model made up of two compartments simulates the irregular bursting ofthe LP neuron. The location of the conductances is shown on the cell diagram and their electronic circuit counterpart is shown onthe right. Changes in burst dynamics can be produced by raising the cytoplasmic calcium concentration (middle panel). The lowerpanel shows a comparison between the burst activity produced by the model (right) and a biological neuron (left).


3.2. Synchronization intwo inhibitory coupled model H-R neurons

The simplest minimal circuit that we studiedconsisted of two identical synaptically coupledneurons. We started by looking at inhibitorysynapses because they represent the predominantsynaptic type in the stomatogastric system andbecause there has been considerable theoreticalwork on reciprocal inhibition as the basis foralternate bursting in rhythmic motor systems. Inaddition, inhibitory synapses appear to play amajor role in coordinating the timing of bursts inrhythmic systems and in regulating the dynamicsof coupled chaotic neurons. We characterized cou-pled H-R neurons as having a threshold and aconstant resting potential and added to the equa-tion describing the membrane potential x1 of oneneuron a synaptic current associated with the ac-tion of another neuron with potential x2 to give:

Is= − [o+h(t)][x1(t)+Vc ]u [x2(t−tc)−X ]

where o is the strength of the coupling and h(t) asmall zero mean noise term. The model showsseveral different regions of synchronization de-pending on the values of synaptic strength andsynaptic delay. When both are small, there is aregion of complete in-phase synchronization. Asthe strength is increased, the neurons become lesssynchronized until they fire out-of-phase with oneanother. The inhibitory coupling reliably regu-larized the behavior of the individually chaoticneurons except when the coupling was very weak.

4. Electronic neurons

Using three and four-dimensional Hindmarsh-Rose models, we have constructed low dimen-sional analog electronic neurons (ENs) whoseproperties emulate the membrane voltage charac-teristics of individual stomatogastric neurons [16].The ENs are simple analog circuits which integratethe four-dimensional differential equations repre-senting fast and slow subcellular mechanisms thatproduce the characteristic regular/chaotic spiking-bursting behavior of these cells. A schematic dia-gram for the EN is shown in figure 8. Our strategyfor building an analog device instead of usingnumerical integration of the mathematical modelon a PC or DSP board was due to the fact thatdigital integration of the H-H equations is a slowprocedure associated with the two or three differ-ent time scales in the model We also plan to

Figure 8. Block diagram of the four-dimensional HR+Ca2+

neuron used in our experiments. These neurons were based onthe equations shown in the text and were designed to replicatethe behavior of individual isolated neurons from the lobsterSTG. In our experiments the electronic neurons were coupledto each other electrically as well as via an electronic implemen-tation of inhibitory and excitatory chemical synapses.

eventually construct real-time networks of the en-tire pyloric CPG and for a large number of neu-rons and for this, analog implementation is anecessity.

4.1. Beha6ior of the single electronic neuron

As mentioned previously, the equation for w(t)represents an even slower process than z(t) so thatnBm�1 and is included because a slow processsuch as the calcium exchange between intracellularstores and the cytoplasm was found to be requiredin Hodgkin-Huxley modeling [7] to fully reproducethe observed chaotic oscillations of STG neurons.Both the three-dimensional and four-dimensionalmodels have regions of chaotic behavior, but thefour-dimensional model has much larger regions inparameter space where chaos occurs presumablyfor the many of the same reasons that calciumdynamics give rise to chaos in HH modeling


(figure 7). The main parameters we used in con-trolling the modes of spiking and bursting activityof the model are the DC external current I and thetime constants m and n of the slow variables (figure9). The Lyapunov exponents lI for both the 3-Dand 4-D ENs have positive exponents in bothcases indicating conclusively that the behavior ofthe model was chaotic.

4.2. Synaptic connections between ENS

In order to study the functional relationshipsbetween ENs, we also developed circuits to simu-late excitatory and inhibitory synaptic connectionsas well as ohmic electrical connections. Electricalsynapses were implemented by injecting a currentinto one of the neurons proportional to the voltage

Figure 9. The electrical behavior of an electronic neuron. A shows a simultaneous time series of the computed state variables x(membrane potential), y (fast conductance), z (slow conductance) and w (cytosolic calcium concentration). B shows the simulatedactivated of two ENs coupled by synaptic inhibition and their phase portraits.


Figure 10. Positive electrical coupling of two chaotic ENs. Characteristic time series of the membrane potentials x1(t), x2(t) [A–F] assynaptic conductance GE is varied. Phase portraits (a1–f1) are shown on the right unfiltered and after low pass filtering (a2–f2). (A)Intermittent out-of-phase activity, (B) nearly independent chaotic spiking-bursting pattern, (C) chaotic oscillations with most burstssynchronized, (D) periodic oscillations with partial synchronization of the ENs, spikes not synchronized, (E) periodic oscillations withcomplete synchronization of the ENs, (F) chaotic but completely synchronized oscillations.

difference between the two membrane potentialsand injecting the same current with the oppositepolarity into the other neuron.

Although chemical synapses could be imple-mented in analog circuitry, we found it more use-ful to use software so that we could investigate the

Figure 11. Inhibitory chemical coupling between two chaotic ENs. Characteristic time series of the membrane potentials (A–F). Phaseportraits (a1 to e1) and phase portraits after 20 Hz low-pass filtering (a2 to e2). (A) Chaotic oscillations with all hyperpolarized regionsout-of-phase, (B) periodic pattern with hyperpolarizing regions out-of-phase and some burst superposition, (C) chaotic oscillations,(D) periodic out-of-phase bursting behavior with some burst superposition, (E) periodic out-of-phase spiking-bursting behavior, (F)chaotic out-of-phase spiking-bursting pattern.


role of the synaptic time constant ts. We found theresults of the software and hardware models wereidentical for fixed time constants. The electronicneurons were able to generate the currents corre-sponding to the graded chemical synapses of STGneurons in real time described by first order kinet-ics. The synaptic reversal potentials were selectedso that the currents injected into the postsynapticENs were always negative for inhibitory synapsesand positive for excitatory synapses.

4.3. Electrical coupling between two ENs

When the coupling strength GE:0.0, the twoneurons are effectively uncoupled and display in-dependent chaotic oscillations. However as thestrength of the coupling increases, some regions ofsynchronized bursting begins to appear (figure 10)including regions where intermittent anti-phasebursting can be observed. As the coupling is madestronger the bursting activity becomes regular andgoes from a region of partial synchronization to aregion of total synchronization. Between values of0.8 and 1, there is total synchronization in thespiking bursting activity and the oscillations arechaotic. For negative electrotonic coupling, some-thing that does not occur in nature, the oscillationsare mostly chaotic and the bursts occur in anti-phase. As with experiments on biological neurons,there is a sharp phase transition between nearlysynchronous and fully synchronous behavior [15].

4.4. Excitatory chemical-type synapsesbetween ENs

When two uncoupled ENs were adjusted to giveindependent chaotic oscillations, they became reg-ularized as they transitioned between simulatedcurrents 0BGCB100 nS. At very low currents (10nS) the chaotic activity was replaced by a behaviorin which most of the bursts were synchronized butthe oscillations were still chaotic. At 100 nS all thebursts became synchronized and the activity be-comes periodic. At GC\100 nS the bursts remainsynchronized and get longer but there are nolonger any spikes during the final part of thebursts.

4.5. Inhibitory chemical-type synapsesbetween ENs

ENs connected with reciprocal chemical in-hibitory synapses mimic the most common form ofcommunication between pattern forming neurons

in CPGs. We suggested previously that inhibitorychemical coupling could lead to regularization ofchaotic oscillations in individual neurons [2]. Wefound that with small inhibitory currents the ENoscillations were still chaotic and all of the hyper-polarizing regions of the membrane voltages werein anti-phase (figure 11). As the strength of theinhibitory synapses were increased, the oscillationsbecame alternately periodic and chaotic until atvery high synaptic currents long out-of-phasebursts were observed along with chaotic oscilla-tions. While these results are consistent with whatwe have observed in biological neurons, no directcomparison of these simulations with biologicalneurons has been made. When very strong inhibi-

Figure 12. The pyloric pacemaker group, the AB and PDsalong with the VD neuron are able to sustain bursting after theglutamatergic synapses are blocked with picrotoxin (A) how-ever when the AB is photoinactivated (B) the remaining neu-rons become tonic initially and after about one hour resumebursting activity.


Figure 13. A hybrid circuit in which a EN that is firing like AB is inserted into a biological circuit in which AB has been killed. Priorto the vertical line in the center, the current between the EN and the PDs is shut off and the pyloric rhythm is tonic as shown in thePD intracellular trace and extracellular LVN trace. After the positive conductance coupling is turned on as shown in the top trace,the pyloric pattern is reinstituted.

tion is delivered via a dynamic clamp, Ih is acti-vated sufficiently to keep the oscillations regular(unpublished).

4.6. Hybrid silicon-biological circuits

One way to test the verisimilitude of our ENswas to interface them with single biological neu-rons or with the entire pyloric circuit. It wouldthen be possible in principle to see if the EN couldsubstitute for the real neuron under a variety ofperturbations as well as in steady state conditions.The pyloric system is exceptionally well-suited forthese experiments. The fourteen neuron pyloricnetwork has one interneuron, the AB, that is

connected to the two PD neurons via electrotonicjunctions (see figure 1). Together these three neu-rons appear to form the pacemaker for the pyloricsystem, setting the frequency and periodicity of thepyloric pattern. The AB is the strongest neuron ofthe three in this respect and because it is robustand highly periodic it has been difficult to manipu-late the rhythm when it is intact.

The AB can be removed from the pyloric circuitby photoinactivation [14] and in some cases gluta-matergic transmission from other pyloric neuronsonto the PDs was blocked with picrotoxin (7.5mM). A three-dimensional EN was connected tothe two PD neurons via an artificial electricalsynapse [22] which allowed us to dynamically in-


teract with the biological neurons rather than sim-ply providing them with current commandsthrough a microelectrode.

When the AB was removed from a reducedcircuit leaving only the PDs still connected to theVD neuron, the normal oscillatory pattern wasimmediately disrupted and the PD and VD firedtonically and irregularly (figure 12). If the EN wasadjusted to fire in bursts similar to those observedin the AB and then connected to the PDs by anartificial electrical synapse, the PD now fired syn-chronously with the EN (figure 13). When the ENis also made to fire irregularly and then connectedto the chaotically firing PDs, the ENs and the PDswent immediately into synchronous in-phasebursting/firing activity (figure 14A).

Similarly, if the electrical coupling was madenegative, bursting/firing activity also ensued butnow was out-of-phase (figure 14B). If the gluta-matergic activity was blocked, the whole rhythmbecame severely disrupted immediately after theAB pacemaker neuron was photoinactivated. Afterabout 1 h, the pyloric rhythm again become regu-larized although not as strongly as before. Replac-

ing the destroyed AB with an artificial AB couldrescue the circuit and the entire rhythm returned(figure 15). Experiments were also carried out us-ing a 4-D EN with similar results [16].

Previous efforts using hybrid circuits have beenperformed using a digital version of a H-H modelwhich ran on a DSP board of a PC and was usedto replace PD, LP and PY neurons. This hybridcircuit demonstrated that many aspects of the py-loric rhythm could be accurately reproduced [17].This laboratory subsequently developed VLSIdevices for implementing HH type models and hasused them in hybrid circuits with results similar toours but we have found that thus far such complexneurons are not needed in order to retain theoutput properties and dynamical features of thenetwork.

Modeling with hybrid circuits is an interestingway of testing how well an electronic neuron caninteract with a small numbers of neurons in awell-characterized biological CPG. So far, our sili-con neuron having only four degrees of freedomappears to be capable of behaving indistinguish-ably from its biological counterparts. We therefore

Figure 14. Hybrid circuit in which an irregular EN is coupled to an irregular biological PD neuron via positive (A) and negative (B)conductance coupling. The coupling produces an in-phase and out-of-phase periodic bursting/spiking pattern in the two neurons.


Figure 15. Same as figure 13 but with negative conductance coupling which can also restart the pyloric rhythm. Note the EN and PDare out-of-phase in this non-biological condition.

appear to have captured enough of the salientfeatures of real neurons to allow their substitutioninto a small biological network or we have not yettested the electronic neuron hard enough to exposeits frailties. We find it remarkable however thatour ENs are not only able to drive the biologicalneurons but are also influenced by them as welli.e., they actually become part of the CPGcircuitry.

5. Conclusions

– The individual neurons in an invertebrate CPGcircuit have membrane potential behaviors thatvary from periodic to irregular but for non-tran-

sient behaviors, they act as low dimensional oscil-lators.– Irregularly firing neurons coupled to each othervia inhibitory synapses tend to regularize betterthan neurons with excitatory coupling.– The modified Hindmarsh-Rose model appearsto be able to capture the essential dynamical fea-tures of stomatogastric neurons and can serve asthe basis for constructing analog electronic neu-rons which operate in real time.– Such simple electronic neurons, when interfacedwith biological circuits, behave at the level ofmembrane potentials, like neurons.– The Hodgkin-Huxley model with added Cadynamics (1) displays biologically realistic chaoticregimes and (2) can be used to explore the un-derlying cellular mechanisms which produce chaos


and (3) is better than the H-R model in termsof linking experimental data with theoreticalresults.

We now know that chaotic neurons are muchmore reliable and diverse in their properties thanwas once believed. We have shown in our experi-ments that a very small collection of chaotic neu-rons can form an adaptive and flexible patterngenerator which is very sensitive to sensory andcentral sources of input.

How does nature use chaos and how do smallneural networks like invertebrate CPGs withchaotic neurons produce such beautifully regularpatterns? We believe we have begun to answer thatquestion with the experiments presented here. Ofcourse a chaotic neuron may not actually have afunction as such and may be chaotic simply be-cause it is itself a complex nonlinear system. Andquestions remains — do CPGs use chaos to theiradvantage or do they try to minimize it? Dostochastic processes influence the dynamic behav-ior of nerve cells? Nonlinear dynamics gives neu-rons the flexibility and information processingabilities that allow them to perform new kinds ofactivities. The limit cycle attractors of highly peri-odic oscillators are rigid while unstable trajectoriesuse more degrees of freedom — needed if flexibil-ity and reliability are to be achieved. It appearsthat nature has opted for the more flexibleapproach.

References

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