Directory UMM :Data Elmu:jurnal:B:Biosystems:Vol58.Issue1-3.2000:

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Near Poisson-type firing produced by concurrent excitation

and inhibition

Chris Christodoulou

_{*, Guido Bugmann}

a_{School of Computer Science & Information Systems,}_{Birkbeck College,}_Uni

6ersity of London,Malet Street, London WC1E7HX,UK

b_{School of Computing,}_Uni

6ersity of Plymouth,Drake Circus,Plymouth PL4 8AA,UK

Abstract

The effect of inhibition on the firing variability is examined in this paper using the biologically-inspired temporal noisy-leaky integrator (TNLI) neuron model. The TNLI incorporates hyperpolarising inhibition with negative current pulses of controlled shapes and it also separates dendritic from somatic integration. The firing variability is observed by looking at the coefficient of variation (CV) (standard deviation/mean interspike interval) as a function of the mean

interspike interval of firing (DtM) and by comparing the results with the theoretical curve for random spike trains, as

well as looking at the interspike interval (ISI) histogram distributions. The results show that with 80% inhibition, firing at high rates (up to 200 Hz) is nearly consistent with a Poisson-type variability, which complies with the analysis of cortical neuron firing recordings by Softky and Koch [1993, J. Neurosci. 13(1) 334 – 530]. We also demonstrate that the mechanism by which inhibition increases theCVvalues is by introducing more short intervals in the firing pattern

as indicated by a small initial hump at the beginning of the ISI histogram distribution. The use of stochastic inputs and the separation of the dendritic and somatic integration which we model in TNLI, also affect the high firing, near Poisson-type (explained in the paper) variability produced. We have also found that partial dendritic reset increases slightly the firing variability especially at short ISIs. © 2000 Elsevier Science Ireland Ltd. All rights reserved.

Keywords:High firing variability; Inhibition; Coefficient of variation; Temporal noisy-leaky integrator

www.elsevier.com/locate/biosystems

1. Review of the problem: determinants of the highly variable neuronal firing and the neural code

The controversy surrounding the issue of the neural code and of the determinants of the highly variable neuronal firing has recently been revi-talised by Softky and Koch (1993). These authors demonstrated that the classical notion of a realis-tic neuron, i.e. being of a leaky-integrator type, failed in reproducing the high firing variability * Corresponding author. Present address: Department of

Electronic Engineering, King’s College, University of London, Strand, London, WC2R 2LS, UK. Tel.: +44-20-7631-6718; fax: +44-20-7631-6727.

E-mail addresses: [email protected] (C. Christodoulou), [email protected] (G. Bugmann).

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they observed in cortical cells at high firing rates. Using a simple leaky integrate-and-fire (I&F) neuron and also a detailed compartmental model, they could only obtain high variability at low firing rates and concluded that the neural code is based on temporal precision of input spike trains, i.e. neurons behave as coincidence detectors rather than leaky integrators. Shadlen and Newsome (1994), used a random walk model and by appropriate balancing of excitation and inhibition on a single cell, they produced highly irregular firing. They concluded that the neural code is based on rate encoding rather than pre-cise processing of coincident presynaptic events. Bell et al. (1995), who supported the coincidence detection principle, produced high irregular firing using a single compartment Hodgkin and Huxley (1952) model (abbreviated H&H) with balanced excitation and inhibition (with the ‘balance point’ near the threshold in contrast to Shadlen and Newsome, 1994), in addition to weak potas-sium current repolarisation (which corresponds to the degree of somatic reset) and fast effective membrane time constants. Ko¨nig et al. (1996) supported the coincidence detection principle as a possible mode for neural operation by disput-ing Shadlen and Newsome’s (1994) finddisput-ings; they questioned in particular the biological realism of their assumptions, namely that there is an exact balance between excitatory and inhibitory inputs and the high rate of input signals. The assump-tion of how balanced excitaassump-tion and inhibiassump-tion is brought about naturally in model networks has also been studied by Van Vreeswijk and Som-polinsky (1996, 1998) and Amit and Brunel (1997). Shadlen and Newsome (1998) reiterate their previous findings by reinforcing both of their questionable assumptions with experimental evidence.

In an attempt to model high irregularity, we have demonstrated (Bugmann et al., 1997), using a simple leaky integrator model withpartial reset on the somatic membrane potential, that irregu-lar firing can be produced at high firing rates resulting from both temporal integration of ran-dom excitatory post-synaptic potentials (EPSPs) and current fluctuation detection (partial somatic reset was also examined previously by La´nsky´

and Smith, 1989; La´nsky´ and Musila, 1991). We have also showed that the partial reset is a pow-erful parameter to control the gain of the neu-ron. The results of Softky and Koch (1993) have also been reproduced by Lin et al. (1998), by using precise stochastic coupling in a network of I&F neurons arranged in a one-dimensional ring topology.

Feng and Brown (1998) used an I&F model and showed that the CV (coefficient of variation

— measure of spike train irregularity defined as the standard deviation divided by the mean ISI) of the output firing is an increasing function of the length of the distribution of the input inter-arrival times and the degree of balance between excitation and inhibition (r). They also showed that there is a range of values ofrthatCVvalues

between 0.5 and 1 can be achieved (which is considered to be the physiological range) and this range excludes exact balancing between exci-tation and inhibition. Moreover these authors demonstrated elsewhere (Feng and Brown, 1999) that CV values [0.5, 1] can also be obtained

using a leaky I&F model (Stein’s model) with and without reversal potentials when the attrac-tor of the deterministic part of the dynamics is below the threshold and firing results from ran-dom fluctuations. In another study Brown et al. (1999) examined the variability of the H&H and FitzHugh-Nagumo neurons with random synap-tic input and showed that CV[0.5,1] can be

ob-tained which are not dependent on the inhibitory input level.

2. Neuron model used: the temporal noisy-leaky integrator

For this study we used the TNLI neuron model (Christodoulou et al., 1992, 1994), which is a simple, biologically inspired and hardware realis-able computational model. Fig. 1 shows an ana-logue hardware outline of the TNLI using a pRAM (probabilistic RAM, Clarkson et al., 1992) at each input and a H&H equivalent circuit for a leaky cell membrane (implementational de-tails of the TNLI digital hardware realisation can be found in Christodoulou et al., 1992 and its theoretical analysis in the Appendix A). The

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1-pRAMs in the TNLI model the stochastic and spontaneous neurotransmitter release by the synapses of real neurons. The 0-pRAMs shown in the model are used in the simulations to produce random spike input trains from other neurons of controlled mean input frequency, according to their probability p. The postsynaptic response (PSR) generators (Fig. 1), model the effects of dendritic propagation of the postsynaptic poten-tials and in particular their temporal summation (Nicholls et al., 1992). The presynaptic transmit-ter release creates an ion-specific conductance change in the postsynaptic neuron which in the TNLI we approximate with an inward or out-ward current flow model (see Eq. (A2), Appendix A). The separation of dendritic and somatic inte-gration make the current-based model approxi-mation necessary, because a current input is needed to the leaky integrator circuit following in the model, which is theacti6esingle-compartment

representing the somatic membrane. We have therefore voltage as output of that circuit, repre-senting the somatic membrane potential. For ev-ery spike generated by the pRAMs, the PSR generators produce postsynaptic current re-sponses PSRij(t) (i.e. a postsynaptic response at

neuron icaused by an input spike at time tfrom

input neuron j), of controlled shapes, shown in Fig. 1 (at inputs n and n+m), which can either be excitatory (excitatory postsynaptic currents, EPSCs) or inhibitory (inhibitory postsynaptic currents, IPSCs). Such EPSCs and IPSCs ex-tended in time, have been used previously in the form of an alpha function (see Walmsley and Stuklis, 1989 and references therein). In the TNLI, these particular ramp shapes chosen for the PSRs are an approximation of alpha func-tions in the form of linear splines that can easily be implemented in digital hardware.

the TNLI neuron fires. It then waits for an abso-lute refractory period (tR) and fires again if the

membrane potential is above the threshold after the refractory period elapses. Therefore, the max-imum firing rate of the TNLI is given by 1/tR. In

this model the integration of inputs continues during the refractory time, but without having the value of the membrane potential compared with the firing threshold during that time. De-pending on the application for which the TNLI is being used, the membrane potential, i.e. the po-tential of the capacitor, can be completely dis-charged or reset whenever the neuron fires (Christodoulou et al., 1994) (as in Lapique, 1907) or not reset at all (Christodoulou et al., 1992) (as in Bressloff and Taylor, 1991) or partially reset (as in Bugmann et al., 1997). For the simulations Fig. 1. Analogue hardware outline of the TNLI neuron model.

The dotted line boxes indicate the corresponding parts of the real neuron which the TNLI modules are inspired by. At inputsnandn+m, the postsynaptic current response shapes utilised are shown (EPSCs and IPSCs, respectively) where:td:

synaptic delay time;tp: peak period time;dr: rise time;df: fall

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described in this paper, total somatic reset was applied on the somatic potential of the TNLI.

Therefore, the main differences of the TNLI from other models are: (i) the separation of den-dritic and somatic integration (similar to the mod-els of Kohn, 1989 and Rospars and La´nsky´, 1993); (ii) the modelling of the temporal summation of the PSPs in the dendrites; and (iii) the use of stochastic synapses (represented by the pRAMs).

3. Simulation data and results

The parameter values used for the postsynaptic current responses (Fig. 1) are:td=5 ms,dr=df=

5 ms, tp=10 ms, h=5 pA. For simplicity, the

inhibitory currents have an equal but opposite magnitude to the excitatory ones (−5 pA). The other TNLI parameters are: tR=2 ms, Vth=15

mV, R=166 MV, C=60 pF (giving a realistic membrane time constant t=RC#10 ms). The simulation time step used was Dt=1 ms and the system was left to operate for T=10 000 ms. At the TNLI inputs, random spike trains of con-trolled mean frequency (fj) were utilised with

fj=p/Dt, where p is the 0-pRAM probability

value. These random spike trains were unaffected by the 1-pRAM action in the current simulations. fjwas the same for both excitatory and inhibitory

inputs.

Results were taken with 100 excitatory PSR generators and 0, 40, 80 and 95 inhibitory ones (denoting the number of excitatory and inhibitory synaptic inputs). Fig. 2 shows theCVas a function

of DtM (mean interspike interval of firing) while

the number of inhibitory inputs was increased. Full dendritic reset has been applied for these results. The full line shows the theoretical curve for a random spike train with discrete time steps given by:

CV=

'

DtM−tR

DtM

(see Bugmann, 1995; Bugmann et al., 1997). If the simulated firing ISIs are poissonian, then theirCV

versus DtM curve should follow this theoretical

curve. TheCVvalues obtained with 100 excitatory

inputs and 80 inhibitory inputs (100 ex/80 inh, Fig.

Fig. 2. Coefficient of variation (CV) versus mean interpike

interval (DtM) showing the firing variability obtained with the

TNLI neuron at different levels of inhibition. These results are taken with full dendritic reset. For the rest of the details and parameter values, see text.

inhibition levels, we can see that with 80% inhibi-tion when CV=0.870, the distribution follows a

Poisson tail (exponential decay). The small initial hump at the beginning of the distribution is due to the presence of clusters of spikes at short intervals. Moreover it can also be observed from Fig. 2 that high variability and near Poisson distributions (with 80% inhibitions) are obtained at the cost of CV’s slightly larger than one at long intervals

(around DtMof 30 ms). By near Poisson firing we

mean that theCVversusDtMcurve is very close to

Fig. 3. Interspike Interval histogram distributions for DtM=

15 ms (with dendritic reset) for different inhibition levels. T indicates the total time the system was left to operate.

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Fig. 4. ISI histogram distributions for different mean ISI lengths for the case of 80% inhibition with dendritic reset. The system is left to operate forT=50 000 ms.

4. Discussion and conclusions

Shadlen and Newsome (1994, 1998) showed that exact balancing of excitation and inhibition can produce highly irregular firing and proved that by looking at both the range of CVvalues of

the output firing and the interspike interval distri-butions. Our results suggest that only around 80% inhibition on concurrent excitation is needed to produce highly irregular firing and not exact bal-ance. The results of Feng and Brown (1998, 1999) and Brown et al. (1999) (who also examined the effect of inhibition on the firing variability — see review above) do not give a clear and complete answer to the problem posed by Softky and Koch (1993) for two reasons: (i) it is not clarified whether theCVvalues obtained are for high firing

rates; and (ii) only the CV statistic for assessing

high variability is utilised, which on its own is not a reliable indicator, since CVvalues [0.5, 1] are

not necessarily equivalent with Poisson statistics. It has to be noted that the way the above authors produce inhibition in their models is different from ours: in Shadlen and Newsome (1994, 1998) both excitatory and inhibitory inputs to the neu-ron are modelled as simple time-series as in Feng and Brown (1998, 1999) and Brown et al. (1999), but in the latter ones the input can follow Gaus-sian, Poisson or Pareto distributions. In Feng and Brown (1999) the number of excitatory inputs is equal to the inhibitory inputs and the authors change the rate of inhibitory firing. In addition no one of the above authors separate in the their model the dendritic and somatic integration as we the theoretical one and the ISI histogram

distribu-tions are nearly exponentially distributed. There-fore, we can deduce that with 80% inhibition on concurrent excitation, random firing spike trains can be produced at high firing rates as observed in cortical neurons (Softky and Koch, 1993). The use of stochastic inputs and the separation of the dendritic and somatic integration which we model in TNLI, also affect the high firing, near Poisson-type variability produced. The effect of the width of the postsynaptic responses is investigated in Bugmann (1995), where alpha functions were used for postsynaptic currents.

Fig. 4 shows the ISI histogram distributions for two further mean ISI values (10 and 20 ms, with CV’s ofCV=0.742 and 0.954, respectively) on top

of the ISI distribution for DtM=15 ms for the

case of 80% inhibition. As it can be observed all the distributions have a Poisson tail with a hump at the beginning indicating the presence of clusters of spikes at short intervals. The number of short intervals are greater for DtM=10 ms while the

tails of all three distributions remain the same. The fluctuations in the distribution with_DtM=10

ms at long intervals are due to noise caused by the small number of intervals present at that range.

We have also examined the effect of partial dendritic reset on the firing variability for the case of 80% inhibition (see Fig. 5) and found that it slightly increases the firing variability.

Fig. 5. Coefficient of variation for 80% inhibition at different levels of dendritic reset.

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do here with the TNLI. To the best of our knowl-edge, no one has carried out an analysis where the firing at different levels of inhibition has been compared with the theoretical Poisson-type firing resembling the findings of the analysis of cortical neuron recordings by Softky and Koch (1993), as we do here.

The effect of partial dendritic reset is to very slightly increase the CV values (see Fig. 5). The

increase is noticeable below DtM=10 ms and

minor above it (compared to the major effect of partial somatic reset, Bugmann et al., 1997). Par-tial dendritic reset allows the accumulated postsy-naptic current to remain at higher values which enables the potential, after integration, to be closer to the threshold. This produces more spikes which at short intervals (for the 50% dendritic reset case) increases slightly the irregularity (since the CV curve follows exactly the theoretical one

for high variability), but at long intervals it fa-vours bursting.

Therefore, the main conclusion is that concur-rent excitation and inhibition (with inhibition be-ing at a level of approximately 80% of excitation) can produce the Poisson-type firing observed in cortical neurons using the TNLI model where there is separation of dendritic and somatic inte-gration (with total dendritic and somatic reset) and stochastic input spike trains. This method can be added to the two other mechanisms that have previously been reported for producing Poisson-type firing namely: partial somatic reset of a leaky integrate-and-fire neuron (Bugmann et al., 1997) and precise stochastic coupling in a network of integrate-and-fire neurons arranged in a one-di-mensional ring topology (Lin et al., 1998). Fur-ther studies are needed to determine which of these methods best reflect what is actually hap-pening in the brain that causes Poisson-type firing in cortical neurons. In addition, the increase of theCVvalues towards one is due to the presence

of clusters at short intervals whereas high firing variability and near Poisson distributions are ob-tained at the cost CV’s slightly larger than one at

long intervals. The mechanism therefore by which inhibition increases theCV values is by

introduc-ing more short intervals in the firintroduc-ing pattern. We have also showed that partial dendritic reset

in-creases slightly the firing variability, especially at short ISIs.

Appendix A. Theoretical basis of the TNLI

Most of the leaky integrator models are based on the equation that Hodgkin and Huxley (1952) used to describe the generation of an action po-tential in the giant squid axon (similar equation first used by Lapique, 1907). By linearising that equation, a simplified version for a network of single-compartment leaky integrator neurons with synaptic noise (as used in Bressloff and Taylor, 1991), can be described by the shunting differen-tial equation:

dVi

dt = − Vi(t)

+_%

j"i

Dgij(t)×[Sij−Vi(t)]

(I) (II) (III)

(A1) where:

term (I) is the variation of accumulated charge in neuron i,

term (II) is the membrane leakage current in neuron i (negative term) and

term (III) is the synaptic input current which is excitatory for Sij\0 and inhibitory for SijB0

(where 0 mV is considered to be the resting membrane potential instead of −70 mV). Vi(t) is the membrane potential of theith neuron

at timetandDgijis the increase in conductance at

the synaptic connection between neuron j and neuron i, with membrane reversal potential Sij,

due to the release of chemical neurotransmitters. Ri is the leakage resistance and Ci the somatic

capacitance of the membrane.

The TNLI corresponds to the model described by Eq. (A1) since it consists of a single active compartment representing the soma and perform-ing integration of EPSCs and IPSCs, and it also does not differentiate between different ionic cur-rents. For the hardware TNLI model however, Eq. (A1) is further simplified by assuming that [Sij−Vi(t)] is a constant close to the resting

po-tential so that the synaptic current flow is inde-pendent of the membrane potentialVi(t). This is a

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reasonable approximation since Sij is approx. 3

mV for an EPSP (Coombs et al., 1955), while Vi(t) would always lie between −70 mV (resting

membrane potential) and −55 mV (action poten-tial threshold). Note that for our simulations, the resting membrane potential is shifted to 0 mV for simplicity. Term (III) as a whole, which represents the current flow into the soma, corresponds in our model to the total postsynaptic response current produced by the temporal summation of the post-synaptic current responses each of which is ini-tiated by an input spike. Thus, after the above approximation, the leaky integrator equation for the TNLI becomes:

Ci=

dVi

dt = − Vi(t)

+_%

05tk5T

PSRij(t−tk)

(A2) where PSRij (t−tk) is the postsynaptic current

response caused by an input spike having arrived at time tkfrom input neuronj. The response can

either be excitatory or inhibitory depending on the sign of PSRij. T is the total number of time

steps that the system is left to operate. Discrete time steps are used in the simulations so dtcan be replaced by Dtin Eq. (A2), and the double sum-mation term represents current so we can call it I(t). In the hardware model of the TNLI neuron (Christodoulou et al., 1992) the postsynaptic cur-rent responses are accumulated in a counter where they are multiplied at regular time intervals by a decay rate. The decayed synaptic potential is routed back to the counter via a load input. The termCiDVi(which results after discritisation)

rep-resents the counter contents in the TNLI and can be written as CiVi(t+Dt)−CiVi(t). So with the

above simplifications, Eq. (A2) becomes (after dropping the i notation):

CV(t+Dt)=CV(t)+I(t)×Dt−V(t)

R Dt (A3)

In the hardware model of the TNLI this equation can be realised in two steps: First step:

CV*(t)=CV(t)before+I(t)×Dt (A4)

and the second step:

CV(t+Dt)=a×CV*(t), aB1 (A5) where a is the decay rate with which the initial counter output contents (CV*(t)) are multiplied before they are routed back to the counter via the load input. In other words this decay rate replaces the term [(−V(t)/Ri)×Dt] due to the hardware

structure. The relationship between the decay rate a and the time constant t=RC can be deduced from Eqs. (A3), (A4) and (A5) and is given by: a=1− 1

RCDt (A6)

From Eqs. (A4) and (A5) we deduce that the capacitor is charged according to the equation: V(t+Dt)=a×

V(t)+I(t)×Dt

n

(A7)

When reset is applied to the TNLI, after each firing at time t, V(t+Dt)0 (i.e. the membrane potential resets to 0). In other words, the potential of the soma is reset, but not the accumulated postsynaptic current due to the temporal summa-tion in the dendrites. When partial reset is applied to the soma (Bugmann et al., 1997) then after each firing at time t, V(t+Dt)b×V(t); b is the reset parameter taking values between 0 and 1 (b=0 for the simulations described here).

The firing times in the TNLI neuron i(Tn i_{) are}

determined (as in Bressloff and Taylor, 1991) by the iterative threshold condition:

Tn i_=inf{

tVi(t)]Vth_i;t]Tn−1

+tR} n]1

(A8) whereVth_iis the threshold voltage of neuroniand

n is the time of firing.

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pRAMs in the TNLI model the stochastic and spontaneous neurotransmitter release by the synapses of real neurons. The 0-pRAMs shown in the model are used in the simulations to produce random spike input trains from other neurons of controlled mean input frequency, according to their probability p. The postsynaptic response (PSR) generators (Fig. 1), model the effects of dendritic propagation of the postsynaptic poten-tials and in particular their temporal summation

(Nicholls et al., 1992). The presynaptic transmit-ter release creates an ion-specific conductance change in the postsynaptic neuron which in the TNLI we approximate with an inward or out-ward current flow model (see Eq. (A2), Appendix A). The separation of dendritic and somatic inte-gration make the current-based model approxi-mation necessary, because a current input is needed to the leaky integrator circuit following in the model, which is theacti6esingle-compartment

neuron icaused by an input spike at time tfrom

The EPSCs and IPSCs are then summed spa-tio-temporally and the total postsynaptic current response is fed into the RC circuit (Fig. 1). The synaptic saturation that occurs in the real neuron during the temporal summation of the postsynap-tic potentials (Burke and Rudomin, 1977) is not currently modelled in the TNLI, but it could be easily incorporated by applying the methods used in (Bugmann, 1992). The capacitance C and the resistance R represent the somatic leaky mem-brane of real neurons and therefore this circuit models the decay that occurs in the somatic po-tential of the real neuron due to its membrane leak. The capacitance C and the resistance Rare fixed at a suitable value to give the leaky mem-brane time constant (t=RC). For simplicity, the TNLI does not differentiate in its leaky integrator circuit between different ionic currents as may occur in the real neuron. If the potential of the capacitor exceeds a constant threshold (Vth), then the TNLI neuron fires. It then waits for an abso-lute refractory period (tR) and fires again if the membrane potential is above the threshold after the refractory period elapses. Therefore, the max-imum firing rate of the TNLI is given by 1/tR. In this model the integration of inputs continues during the refractory time, but without having the value of the membrane potential compared with the firing threshold during that time. De-pending on the application for which the TNLI is being used, the membrane potential, i.e. the po-tential of the capacitor, can be completely dis-charged or reset whenever the neuron fires (Christodoulou et al., 1994) (as in Lapique, 1907) or not reset at all (Christodoulou et al., 1992) (as in Bressloff and Taylor, 1991) or partially reset (as in Bugmann et al., 1997). For the simulations

Fig. 1. Analogue hardware outline of the TNLI neuron model. The dotted line boxes indicate the corresponding parts of the real neuron which the TNLI modules are inspired by. At inputsnandn+m, the postsynaptic current response shapes utilised are shown (EPSCs and IPSCs, respectively) where:td:

synaptic delay time;tp: peak period time;dr: rise time;df: fall

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described in this paper, total somatic reset was applied on the somatic potential of the TNLI.

3. Simulation data and results

The parameter values used for the postsynaptic current responses (Fig. 1) are:td=5 ms,dr=df= 5 ms, tp=10 ms, h=5 pA. For simplicity, the inhibitory currents have an equal but opposite magnitude to the excitatory ones (−5 pA). The other TNLI parameters are: tR=2 ms, Vth=15

fj=p/Dt, where p is the 0-pRAM probability value. These random spike trains were unaffected by the 1-pRAM action in the current simulations.

fjwas the same for both excitatory and inhibitory inputs.

Results were taken with 100 excitatory PSR generators and 0, 40, 80 and 95 inhibitory ones (denoting the number of excitatory and inhibitory synaptic inputs). Fig. 2 shows theCVas a function

of DtM (mean interspike interval of firing) while the number of inhibitory inputs was increased. Full dendritic reset has been applied for these results. The full line shows the theoretical curve for a random spike train with discrete time steps given by:

CV=

'

DtM−tR

DtM

(see Bugmann, 1995; Bugmann et al., 1997). If the simulated firing ISIs are poissonian, then theirCV

versus DtM curve should follow this theoretical

curve. TheCVvalues obtained with 100 excitatory

inputs and 80 inhibitory inputs (100 ex/80 inh, Fig.

Fig. 2. Coefficient of variation (CV) versus mean interpike

interval (DtM) showing the firing variability obtained with the

TNLI neuron at different levels of inhibition. These results are taken with full dendritic reset. For the rest of the details and parameter values, see text.

2) are very similar to those observed in cortical neurons (see Fig. 9 in Softky and Koch, 1993). By looking also at the ISI histogram distributions for mean ISI DtM of 15 ms (Fig. 3) for the different inhibition levels, we can see that with 80% inhibi-tion when CV=0.870, the distribution follows a Poisson tail (exponential decay). The small initial hump at the beginning of the distribution is due to the presence of clusters of spikes at short intervals. Moreover it can also be observed from Fig. 2 that high variability and near Poisson distributions (with 80% inhibitions) are obtained at the cost of

CV’s slightly larger than one at long intervals

(around DtMof 30 ms). By near Poisson firing we mean that theCVversus_DtMcurve is very close to

Fig. 3. Interspike Interval histogram distributions for DtM=

15 ms (with dendritic reset) for different inhibition levels. T

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Fig. 4. ISI histogram distributions for different mean ISI lengths for the case of 80% inhibition with dendritic reset. The system is left to operate forT=50 000 ms.

4. Discussion and conclusions

Shadlen and Newsome (1994, 1998) showed that exact balancing of excitation and inhibition can produce highly irregular firing and proved that by looking at both the range of CVvalues of

high variability is utilised, which on its own is not a reliable indicator, since CVvalues [0.5, 1] are

Fig. 4 shows the ISI histogram distributions for two further mean ISI values (10 and 20 ms, with

CV’s ofCV=0.742 and 0.954, respectively) on top

of the ISI distribution for DtM=15 ms for the case of 80% inhibition. As it can be observed all the distributions have a Poisson tail with a hump at the beginning indicating the presence of clusters of spikes at short intervals. The number of short intervals are greater for DtM=10 ms while the tails of all three distributions remain the same. The fluctuations in the distribution with_DtM=10 ms at long intervals are due to noise caused by the small number of intervals present at that range.

We have also examined the effect of partial dendritic reset on the firing variability for the case of 80% inhibition (see Fig. 5) and found that it slightly increases the firing variability.

Fig. 5. Coefficient of variation for 80% inhibition at different levels of dendritic reset.

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The effect of partial dendritic reset is to very slightly increase the CV values (see Fig. 5). The increase is noticeable below DtM=10 ms and minor above it (compared to the major effect of partial somatic reset, Bugmann et al., 1997). Par-tial dendritic reset allows the accumulated postsy-naptic current to remain at higher values which enables the potential, after integration, to be closer to the threshold. This produces more spikes which at short intervals (for the 50% dendritic reset case) increases slightly the irregularity (since the CV curve follows exactly the theoretical one

for high variability), but at long intervals it fa-vours bursting.

introduc-ing more short intervals in the firintroduc-ing pattern. We have also showed that partial dendritic reset

in-creases slightly the firing variability, especially at short ISIs.

Appendix A. Theoretical basis of the TNLI Most of the leaky integrator models are based on the equation that Hodgkin and Huxley (1952) used to describe the generation of an action po-tential in the giant squid axon (similar equation first used by Lapique, 1907). By linearising that equation, a simplified version for a network of single-compartment leaky integrator neurons with synaptic noise (as used in Bressloff and Taylor, 1991), can be described by the shunting differen-tial equation:

dVi

dt = − Vi(t)

Ri +j"%i

Dgij(t)×[Sij−Vi(t)]

(I) (II) (III)

(A1) where:

term (I) is the variation of accumulated charge in neuron i,

term (II) is the membrane leakage current in neuron i (negative term) and

term (III) is the synaptic input current which is excitatory for Sij\0 and inhibitory for SijB0 (where 0 mV is considered to be the resting membrane potential instead of −70 mV).

Vi(t) is the membrane potential of theith neuron

at timetandDgijis the increase in conductance at

the synaptic connection between neuron j and neuron i, with membrane reversal potential Sij, due to the release of chemical neurotransmitters.

Ri is the leakage resistance and Ci the somatic capacitance of the membrane.

po-tential so that the synaptic current flow is inde-pendent of the membrane potentialVi(t). This is a

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reasonable approximation since Sij is approx. 3 mV for an EPSP (Coombs et al., 1955), while

Vi(t) would always lie between −70 mV (resting

Ci=

dVi

dt = − Vi(t)

Ri +%j

05tk5T

PSRij(t−tk)

(A2) where PSRij (t−tk) is the postsynaptic current

response caused by an input spike having arrived at time tkfrom input neuronj. The response can either be excitatory or inhibitory depending on the sign of PSRij. T is the total number of time

steps that the system is left to operate. Discrete time steps are used in the simulations so dtcan be replaced by Dtin Eq. (A2), and the double sum-mation term represents current so we can call it

I(t). In the hardware model of the TNLI neuron (Christodoulou et al., 1992) the postsynaptic cur-rent responses are accumulated in a counter where they are multiplied at regular time intervals by a decay rate. The decayed synaptic potential is routed back to the counter via a load input. The termCiDVi(which results after discritisation) rep-resents the counter contents in the TNLI and can be written as CiVi(t+_Dt)−CiVi(t). So with the above simplifications, Eq. (A2) becomes (after dropping the i notation):

CV(t+Dt)=CV(t)+I(t)×Dt−V(t)

R Dt (A3)

In the hardware model of the TNLI this equation can be realised in two steps: First step:

CV*(t)=CV(t)before+I(t)×Dt (A4) and the second step:

a and the time constant t=RC can be deduced from Eqs. (A3), (A4) and (A5) and is given by:

a=1− 1

RCDt (A6)

From Eqs. (A4) and (A5) we deduce that the capacitor is charged according to the equation:

V(t+Dt)=a×

V(t)+I(t)×Dt

n

(A7)

The firing times in the TNLI neuron i(Tn i_{) are}

determined (as in Bressloff and Taylor, 1991) by the iterative threshold condition:

Tn i

=inf{tVi(t)]Vth

i;t]Tn−1 i

+tR} n]1

(A8) whereVth

iis the threshold voltage of neuroniand n is the time of firing.

References

Amit, D.J., Brunel, N., 1997. Global spontaneous activity and local structured (learned) delay activity in cortex. Cereb. Cortex 7, 237 – 252.

Bressloff, P.C., Taylor, J.G., 1991. Discrete time leaky integra-tor network with synaptic noise. Neural Netw. 4, 789 – 801.

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