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 1$\le$i$\le$N 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,

$${J_0\over N}$$ (6.2)

where J₀ is a parameter. For J₀ = 0 all neurons are independent; a value J₀ > 0 (J₀ < 0) implies excitatory (inhibitory) all-to-all 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$\to$$\infty$.

Model neurons are described by formal spiking neurons as introduced in Chapter 4. In the case of leaky integrate-and-fire neurons with

$$\tau_{m}^{} {{\text{d}}\over {\text{d}}t}$$ (6.3)

a homogeneous network implies that all neurons have the same input resistance R, the same membrane time constant $\tau_{m}^{}$, as well as identical threshold and reset values. The input current I_i takes care of both the external drive and synaptic coupling

$$\sum_{{j=1}}^{N} \sum_{f}^{} \alpha$$ (6.4)

Here we have assumed that each input spike generates a postsynaptic current with some generic time course $\alpha$(t - t_j^(f)). The sum on the right-hand side of (6.4) runs over all firing times of all neurons. Because of the homogeneous all-to-all 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,

$$\int_{0}^{\infty} \alpha$$ (6.5)

which is independent of the neuronal index i. As an aside we note that for conductance-based 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 integrate-and-fire 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

$$\eta \hat{{t}}_{i}^{} \hat{{t}}_{i}^{}$$ (6.6)

where $\hat{{t}}_{i}^{}$ is the most recent firing time of neuron i. The kernel $\eta$(.) describes the spike and the spike after-potential while

$$\hat{{t}}_{i}^{} \sum_{{j=1}}^{N} \sum_{{f}}^{} {J_0\over N} \epsilon \hat{{t}}_{i}^{} \int_{0}^{\infty} \kappa \hat{{t}}_{i}^{}$$ (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 $\epsilon$ models the response of the neuron to a single presynaptic input spike while $\kappa$ 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 J₀/N. The population theory developed in this chapter is valid for arbitrary response kernels $\epsilon$ and $\kappa$ and for a broad variety of refractory kernels $\eta$. By an appropriate choice of the kernels, we recover the integrate-and-fire model; cf. Chapter 4.2. If we suppress the t - $\hat{{t}}_{i}^{}$ dependency of $\epsilon$ and $\kappa$, we recover the simple model SRM₀ from Chapter 4.2.3.

Similarly to the approach that we used for the total input current of an integrate-and-fire neuron, we can rewrite Eq. (6.7) in terms of the population activity A,

$$\hat{{t}} \int_{0}^{\infty} \epsilon \hat{{t}} \int_{0}^{\infty} \kappa \hat{{t}}$$ (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|$\hat{{t}}$) of a neuron that has fired its last spike at $\hat{{t}}$. Note that we have suppressed the index i, since all neurons that have fired their last spike at $\hat{{t}}$ 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$\to$$\infty$.

In the absence of noise, the next firing time of a spiking neuron i is found from the threshold condition,

$$\vartheta {{\text{d}}\over {\text{d}}t}$$ (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 integrate-and-fire 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 integrate-and-fire neurons with diffusive noise and turn then to Spike Response Model neurons.