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.