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Model of synaptic drive

 

We assume that the synaptic current tex2html_wrap_inline1109 consists of the sum of very brief--essentially instantaneous--individual excitatory postsynaptic currents (EPSCs). This represents a reasonable simplification of the component of the excitatory input to cortical neurons mediated by fast AMPA receptors (which decay with a time constant of 2-3 milliseconds [Bekkers and Stevens, 1990]), but not for the component mediated by the slower NMDA receptor-gated channels.

The synaptic current driving any neuron results from the spike trains of all the other neurons that make synapses onto it. The postsynaptic current depends both on the precise times at which each of the presynaptic neurons fired, and on the response at each synapse to the arrival of a presynaptic action potential. If the response at each synapse is either unreliable or variable in amplitude, then even the arrival of precisely the same spike train at each terminal will fail to produce identical postsynaptic current. In what follows, the exact sequence of action potentials arriving at each of the presynaptic terminals is the ``signal'', and any variability response to repeated trials on which precisely the same sequence is presented represents the ``noise''.

Following the basic quantal model of synaptic transmission [Katz, 1966], we consider two sources of synaptic variability, or noise. The first is that the probability tex2html_wrap_inline1141 that a glutamate-filled vesicle is released following presynaptic activation may be less than unity in the hippocampus [Allen and Stevens, 1994, Rosenmund et al., 1993, Hessler et al., 1993] and the cortex [Castro-Alamancos and Connors, 1997, Stratford et al., 1996]. The second is that the postsynaptic current in response to a vesicle may vary even at single individual terminals [Bekkers and Stevens, 1990]. This quantal variability may arise, for example, from variable amounts of neurotransmitter filling each vesicle [Bekkers and Stevens, 1990]; but the results of the present study do not depend on the mechanism underlying this variability.

The basic model for the postsynaptic current tex2html_wrap_inline1143 driving the neuron is as follows. We assume that the activity in the population of presynaptic neurons j is given by tex2html_wrap_inline1147, where (by analogy with the output tex2html_wrap_inline1137 above) tex2html_wrap_inline1147 is a binary string that is 1 if the neuron fired and 0 otherwise. When an axon fires, the presynaptic terminal releases transmitter with a probability tex2html_wrap_inline1141. If transmitter is released at time t at synapse j, then the postsynaptic amplitude is given by tex2html_wrap_inline1163, which is a random variable that represents the quantal variability. Thus the total postsynaptic current is given by
 equation411
where the summation index j is over the input neurons, the random process tex2html_wrap_inline1167 representing synaptic failures is a binary string that is 1 when transmitter is released and 0 otherwise, and tex2html_wrap_inline1163 is a random variable that determines the quantal size of releases when they occur. The processes tex2html_wrap_inline1175, tex2html_wrap_inline1177, tex2html_wrap_inline1179 and tex2html_wrap_inline1181 are discrete-time, but for notational convenience we will often suppress the time index i.

A single axon may sometimes make multiple synapses onto a postsynaptic target, or a single synapse might have multiple release sites. We use the term functional contact to describe both these situations. Eq. 2 implicitly assumes that each axon has only a single functional contact onto the postsynaptic neuron. We also consider the case where each axon makes tex2html_wrap_inline1185 multiple functional contacts. In this case, the current tex2html_wrap_inline1143 is given by
 equation416
where the summation index k is over functional contacts, each of which is driven by the same sequence of presynaptic action potentials tex2html_wrap_inline1147. In this model, all the terminals k associated with a single presynaptic axon are activated synchronously, but release failures occur at each contact independently.

The Poisson rate tex2html_wrap_inline1195 (impulses/second) at which EPSCs contribute to the postsynaptic current is given in this model by
equation419
where A is the number of afferent axons, tex2html_wrap_inline1185 is the number of functional contacts per axon (assumed to be the same for all axons), tex2html_wrap_inline1201 is the Poisson rate at which each axon fires (assumed to be the same for all axons), and tex2html_wrap_inline1141 is the release probability at each functional contact (assumed to be the same for all contacts). tex2html_wrap_inline1195 determines the average postsynaptic current and thereby the output firing rate R.

In some of the simulations described below (Figs. 3-5), the parameters tex2html_wrap_inline1185 and tex2html_wrap_inline1141 were varied. In order to keep tex2html_wrap_inline1195 fixed under these conditions, any decrease in these parameters was compensated for by a proportional increase in tex2html_wrap_inline1215. For example, if the release probability tex2html_wrap_inline1141 was reduced to 0.5 from 1, tex2html_wrap_inline1201 was increased two-fold.


next up previous
Next: Information rate of spike Up: Theory Previous: Model of spiking

Tony Zador
Fri Nov 28 10:17:14 PST 1997