Subsections

• 6.5.1 Several Populations
  • 6.5.1.1 Example: Stationary states
  
  
• 6.5.2 Spatial Continuum Limit
  • 6.5.2.1 Example: Field Equations for SRM₀ neurons

6.5 Interacting Populations and Continuum Models

In this section we extend the population equations from a single homogeneous population to several populations. We start in Section 6.5.1 with interacting groups of neurons and turn then in Section 6.5.2 to a continuum description.

6.5.1 Several Populations

Let us consider a network consisting of several populations; cf. Fig. 6.14. It is convenient to visualize the neurons as being arranged in spatially separate pools, but this is not necessary. All neurons could, for example, be physically localized in the same column of the visual cortex. Within the column we could define two pools, one for excitatory and one for inhibitory neurons, for example.

Figure 6.14: Several interacting populations of neurons.

We assume that neurons are homogeneous within each pool. The activity of neurons in pool n is

$${1\over N_n} \sum_{{j\in \Gamma_n}}^{} \sum_{f}^{} \delta$$ (6.113)

where N_n is the number of neurons in pool n and $\Gamma_{n}^{}$ denotes the set of neurons that belong to pool n. We assume that each neuron i in pool n receives input from all neurons j in pool m with strength w_ij = J_nm/N_m; cf. Fig. 6.15. The input potential to a neuron i in group $\Gamma_{n}^{}$ is generated by the spikes of all neurons in the network,

$$\hat{{t}}_{i}^{} \sum_{j}^{} \sum_{f}^{} \epsilon \hat{{t}}_{i}^{}$$ =

$$\sum_{m}^{} \int_{0}^{\infty} \epsilon \hat{{t}}_{i}^{} \sum_{{j\in \Gamma_m}}^{} \sum_{f}^{} {\delta(t-t_j^{(f)}-s) \over N_m}$$ = (6.114)

We use Eq. (6.113) to replace the sum on the right-hand side of Eq. (6.114) and obtain

$$\hat{{t}} \sum_{m}^{} \int_{0}^{\infty} \epsilon \hat{{t}}$$ (6.115)

We have dropped the index i since the input potential is the same for all neurons in pool n that have fired their last spike at $\hat{{t}}$. Note that Eq. (6.115) is a straightforward generalization of Eq. (6.8) and could have been `guessed' immediately; external input I^ext could be added as in Eq. (6.8).

In case of several populations, the dynamic equation (6.75) for the population activity is to be applied to each pool activity separately, e.g., for pool n

$$\int_{{-\infty}}^{t} \hat{{t}} \hat{{t}} \hat{{t}}$$ (6.116)

Equation (6.116) looks simple and we may wonder where the interactions between different pools come into play. In fact, pool n is coupled to other populations via the potential h_n which determines the kernel P_n(t | $\hat{{t}}$). For example, with the escape noise model, we have

$$\hat{{t}} \hat{{t}} \vartheta \left\{\vphantom{ \int_{\hat{t}}^t f[u_n(t'\vert\hat{t})-\vartheta] \, {\text{d}}t' \, }\right. \int_{{\hat{t}}}^{t} \hat{{t}} \vartheta \left.\vphantom{ \int_{\hat{t}}^t f[u_n(t'\vert\hat{t})-\vartheta] \, {\text{d}}t' \, }\right\}$$ (6.117)

with u_n(t|$\hat{{t}}$) = $\eta$(t - $\hat{{t}}$) + h_n(t|$\hat{{t}}$), with h_n(t|$\hat{{t}}$) given by (6.115). Eqs. (6.115) - (6.117) determine the dynamics of interacting pools of Spike Response Model neurons with escape noise.

Figure 6.15: All neurons in group $\Gamma_{n}^{}$ are coupled with synaptic efficacy w_ij = J_nn/N_n. Each pair of neurons i, j with the presynaptic j in groups Gamma_m and the postsynaptic neuron i in $\Gamma_{n}^{}$ is coupled via w_ij = J_nm/N_m.

6.5.1.1 Example: Stationary states

The fixed points of the activity in a network consisting of several populations can be found as in Section 6.4. First we determine for each pool the activity as a function of the total input I_m

A_m = g_m(I_m) (6.118)

where g_m is the gain function of neurons in pool m. Then we calculate the total input current to neurons in pool m,

$$\sum_{n}^{}$$ (6.119)

Inserting Eq. (6.119) in (6.118) yields the standard formula of artificial neural networks,

$$\left(\vphantom{\sum_n J_{mn} \, A_n }\right. \sum_{n}^{} \left.\vphantom{\sum_n J_{mn} \, A_n }\right)$$ (6.120)

derived here for interacting populations of neurons.

6.5.2 Spatial Continuum Limit

The physical location of a neuron in a population often reflects the task of a neuron. In the auditory system, for example, neurons are organized along an axis that reflects the neurons' preferred frequency. A neuron at one end of the axis will respond maximally to low-frequency tones; a neuron at the other end to high frequencies. As we move along the axis the preferred frequency changes gradually. For neurons organized along a one-dimensional axis or, more generally in a spatially extended multidimensional network, a description by discrete pools does not seem appropriate. We will indicate in this section that a transition from discrete pools to a continuous population is possible. Here we give a short heuristic motivation of the equations. A thorough derivation along a slightly different line of arguments will be performed in Chapter 9.

To keep the notation simple, we consider a population of neurons that extends along a one-dimensional axis; cf. Fig. 6.16. We assume that the interaction between a pair of neurons i, j depends only on their location x or y on the line. If the location of the presynaptic neuron is y and that of the postsynaptic neuron is x, then w_ij = w(x, y). In order to use Eq. (6.115), we discretize space in segments of size d. The number of neurons in the interval [n d,(n + 1) d] is N_n = $\rho$ d where $\rho$ is the spatial density. Neurons in that interval form the group $\Gamma_{n}^{}$.

Figure 6.16: In a spatially continuous ensemble of neurons, the number of neurons in a segment d is N = $\rho$ d. The efficacy w_ij between two neurons depends on their location. The coupling strength between a presynaptic neuron j at position x_j $\approx$ md and a postsynaptic neuron i at location x_i $\approx$ nd is w_ij $\approx$ w(nd, md ).

We change our notation with respect to Eq.(6.115) and replace the subscript n in h_n and A_n by the spatial position

$$\hat{{t}} \longrightarrow \hat{{t}} \hat{{t}}$$ (6.121)

$$\longrightarrow$$ A_n(t) A(n d, t) = A(x, t) (6.122)

Since the efficacy of a pair of neurons with i $\in$ $\Gamma_{n}^{}$ and j $\in$ $\Gamma_{m}^{}$ is by definition w_ij = J_nm/N_m with N_m = $\rho$ d, we have J_nm = $\rho$ d w(n d, m d ). We use this in Eq. (6.115) and find

$$\hat{{t}} \rho \sum_{m}^{} \int_{0}^{\infty} \epsilon \hat{{t}}$$ (6.123)

For d$\to$ 0, the summation on the right-hand side can be replaced by an integral and we arrive at

$$\hat{{t}} \rho \int \int_{0}^{\infty} \epsilon \hat{{t}}$$ (6.124)

which is the final result. The population activity has the dynamics

$$\int_{{-\infty}}^{t} \hat{{t}} \hat{{t}} \hat{{t}}$$ (6.125)

where P_x is the interval distribution for neurons with input potential h(x, t|$\hat{{t}}$).

If we are interested in stationary states of asynchronous firing, the activity A(y, t) $\equiv$ A₀(y) can be calculated as before with the help of the neuronal gain function g. The result is in analogy to Eqs. (6.118) and (6.120)

$$\left(\vphantom{\rho \,\int w(x,y)\,A_0(y) \, {\text{d}}y}\right. \rho \int \left.\vphantom{\rho \,\int w(x,y)\,A_0(y) \, {\text{d}}y}\right)$$ (6.126)

6.5.2.1 Example: Field Equations for SRM₀ neurons

In the case of SRM₀ neurons, the input potential h does not depend on the last firing time $\hat{{t}}$ so that Eq. (6.124) reduces to

$$\rho \int \int_{0}^{\infty} \epsilon_{0}^{}$$ (6.127)

We assume that the postsynaptic potential can be approximated by an exponential function with time constant $\tau_{m}^{}$, i.e., $\epsilon_{0}^{}$(s) = $\tau_{m}^{{-1}}$ exp(- s/$\tau_{m}^{}$). Just as we did before in Eq. (6.87), we can now transform Eq. (6.127) into a differential equation,

$$\tau_{m}^{} {{\text{d}}h(x,t)\over {\text{d}}t} \rho \int$$ (6.128)

If we make the additional assumption that the activity A changes only slowly over time, we may replace A by its stationary solution, i.e., A(y, t) = g[h(y, t)]. Here g[h(y, t)] is the single neuron firing rate as a function of the total input potential. For constant input current I₀ and normalized input resistance R = 1 we have h₀ = I₀. In this case, we may identify g(h₀) with the gain function g(I₀) of the neuron - and knowing this we have chosen the same symbol g for both functions.

If we insert A(y, t) = g[h(y, t)] in Eq. (6.128), we arrive at an integro-differential equations for the `field' h(x, t)

$$\tau_{m}^{} {{\text{d}}h(x,t)\over {\text{d}}t} \rho \int$$ (6.129)

We refer to Eq. (6.129) as the neuronal field equation (Amari, 1977a; Feldman and Cowan, 1975; Wilson and Cowan, 1973; Ellias and Grossberg, 1975). It will be studied in detail in Chapter 9.