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6.1 Fully Connected Homogeneous Network
Figure 6.1:
Population of neurons (schematic). All neurons
receive the same input
I^{ext}(t) (left)
which results in a time dependent
population activity A(t) (right).

We study a large and homogeneous population of neurons; cf. Fig. 6.1. By homogeneous we mean that all neurons
1iN
are identical and receive the same external input
I_{i}^{ext}(t) = I^{ext}(t). Moreover, in a homogeneous population, the interaction
strength between the neurons is taken to be uniform,
w_{ij} = , 
(6.2) 
where J_{0} is a parameter. For J_{0} = 0 all neurons are independent; a
value J_{0} > 0 (J_{0} < 0) implies excitatory (inhibitory) alltoall coupling.
The interaction strength scales with one over the number of neurons so that
the total synaptic input to each neuron remains finite in the limit of
N.
Model neurons are described by formal spiking neurons as introduced in
Chapter 4. In the case of leaky integrateandfire neurons with
u_{i} =  u_{i} + R I_{i}(t) 
(6.3) 
a homogeneous network implies that all neurons have the same input resistance
R, the same membrane time constant , as well as identical threshold
and reset values. The input current I_{i} takes care of both the external
drive and synaptic coupling
I_{i} = w_{ij}(t  t_{j}^{(f)}) + I^{ext}(t) . 
(6.4) 
Here we have assumed that each input spike generates a postsynaptic current
with some generic time course
(t  t_{j}^{(f)}). The sum on the righthand
side of (6.4) runs over all firing times of all neurons.
Because of the homogeneous alltoall coupling, the total input current is
identical for all neurons. To see this, we insert Eq. (6.2) and use
the definition of the population activity, Eq. (6.1). We find a
total input current,
I(t) = J_{0}(s) A(t  s) ds + I^{ext}(t) , 
(6.5) 
which is independent of the neuronal index i. As an aside we note that for
conductancebased synaptic input, the total input current would depend on the
neuronal membrane potential which is different from one neuron to the next.
Instead of the integrateandfire neuron, we may also use the Spike Response
Model (SRM) as the elementary unit of the population. The membrane potential
of a SRM neuron is of the form
u_{i}(t) = (t  ) + h_{PSP}(t) , 
(6.6) 
where is the most recent firing time of neuron i. The kernel
(.) describes the spike and the spike afterpotential while
h_{PSP}(t) = (t  , t  t_{j}^{(f)}) + (t  , s) I^{ext}(t  s) ds 
(6.7) 
is the postsynaptic potential caused by firings t_{j}^{(f)} of presynaptic neurons
j or by external input
I^{ext}(t). The kernel models the
response of the neuron to a single presynaptic input spike while
is the response to a unit current pulse. In Eq. (6.7) we have already
exploited Eq. (6.2) and replaced the synaptic efficacies w_{ij} by
J_{0}/N. The population theory developed in this chapter is valid for
arbitrary response kernels and and for a broad variety
of refractory kernels . By an appropriate choice of the kernels, we
recover the integrateandfire model; cf. Chapter 4.2. If we
suppress the
t  dependency of and , we recover
the simple model SRM_{0} from Chapter 4.2.3.
Similarly to the approach that we used for the total input current of an
integrateandfire neuron, we can rewrite Eq. (6.7) in terms of the
population activity A,
h_{PSP}(t) = J_{0}(t  , s) A(t  s) ds + (t  , s) I^{ext}(t  s) ds . 
(6.8) 
Thus, given the activity A(t') and the external input
I^{ext} for
t' < t we can determine the potential
h_{PSP}(t) of a neuron that
has fired its last spike at . Note that we have suppressed the index
i, since all neurons that have fired their last spike at have the
same postsynaptic potential
h_{PSP}. As above, this is an
immediate consequence of the assumption of a homogeneous network and does not
require a limit of
N.
In the absence of noise, the next firing time of a spiking neuron i is found
from the threshold condition,
u_{i}(t) = and u_{i} > 0 . 
(6.9) 
In the presence of noise, the next firing time of a given neuron i cannot be
predicted in a deterministic fashion. In the case of integrateandfire
neurons with diffusive noise (stochastic spike arrival), a large noise level
leads to a broad distribution of the membrane potential and indirectly to a
large distribution of interspike intervals; cf. Chapter 5. In
the case of spiking neurons with escape noise (noisy threshold), firing occurs
probabilistically which results in a similar large distribution of interspike
intervals. In the following sections, we formulate population equations for
various types of spiking neuron and different types of noise. We start in the
next section with a population of integrateandfire neurons with diffusive
noise and turn then to Spike Response Model neurons.
Next: 6.2 Density Equations
Up: 6. Population Equations
Previous: 6. Population Equations
Gerstner and Kistler
Spiking Neuron Models. Single Neurons, Populations, Plasticity
Cambridge University Press, 2002
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