Neuron model used: the temporal noisy-leaky integrator

they observed in cortical cells at high firing rates. Using a simple leaky integrate-and-fire IF 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 HH 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 findings; 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 excitation and inhibition 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 with partial 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 IF neurons arranged in a one-dimensional ring topology. Feng and Brown 1998 used an IF model and showed that the C V 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 of r that C V values 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 C V values [0.5, 1] can also be obtained using a leaky IF 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 HH and FitzHugh-Nagumo neurons with random synap- tic input and showed that C V [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 HH 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 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 the acti6e single-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 PSR ij t i.e. a postsynaptic response at neuron i caused by an input spike at time t from 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 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 R are 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 V th , then the TNLI neuron fires. It then waits for an abso- lute refractory period t R 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 1t R . 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 inputs n and n + m, the postsynaptic current response shapes utilised are shown EPSCs and IPSCs, respectively where: t d : synaptic delay time; t p : peak period time; d r : rise time; d f : fall time; and h: postsynaptic peak current. 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