We assume that the synaptic current 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 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 driving the
neuron is as follows. We assume that the activity in the population of
presynaptic neurons j is given by , where (by analogy with
the output above) 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 . If
transmitter is released at time t at synapse j, then the
postsynaptic amplitude is given by , which is a random
variable that represents the quantal variability. Thus the total
postsynaptic current is given by
where the summation index j is over the input neurons, the random process representing synaptic failures is a binary string that is 1 when transmitter is released and 0 otherwise, and is a random variable that determines the quantal size of releases when they occur. The processes , , and 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 multiple functional
contacts. In this case, the current is given by
where the summation index k is over functional contacts, each of which is driven by the same sequence of presynaptic action potentials . 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 (impulses/second) at which EPSCs contribute
to the postsynaptic current is given in this model by
where A is the number of afferent axons, is the number of functional contacts per axon (assumed to be the same for all axons), is the Poisson rate at which each axon fires (assumed to be the same for all axons), and is the release probability at each functional contact (assumed to be the same for all contacts). determines the average postsynaptic current and thereby the output firing rate R.
In some of the simulations described below (Figs. 3-5), the parameters and were varied. In order to keep fixed under these conditions, any decrease in these parameters was compensated for by a proportional increase in . For example, if the release probability was reduced to 0.5 from 1, was increased two-fold.