# 12.2 Identical Neurons: A Mathematical Abstraction

As discussed in the previous section, there exist many brain regions where neurons are organized in populations of cells with similar properties. Prominent examples are columns in the somatosensory and visual cortex (354; 230) and pools of motor neurons (249). Given the large number of neurons within such a column or pool it is sensible to describe the mean activity of the neuronal population rather than the spiking of individual neurons.

The definition of population activity has already been introduced before, but is repeated here for convenience. In a population of $N$ neurons, we calculate the proportion of active neurons by counting the number of spikes $n_{\rm act}(t;t+\Delta t)$ in a small time interval $\Delta t$ and dividing by $N$. Further division by $\Delta t$ yields the population activity

 $A(t)={\rm lim}_{\Delta t\to 0}\,{1\over\Delta t}\,{n_{\rm act}(t;t+\Delta t)% \over N}={1\over N}\sum_{j=1}^{N}\sum_{f}\delta(t-t_{j}^{(f)})\,,$ (12.1)

where $\delta$ denotes the Dirac $\delta$ function. The double sum runs over all firing times $t_{j}^{(f)}$ of all neurons in the population. In other words the activity $A$ is defined by a population average. Even though the activity has units of a rate and indeed is often called the population rate, the population activity is quite different from the concept of a mean firing rate defined by temporal averaging in a single neuron; cf. Section 7.2 in Chapter 7.

Theories of population dynamics, sometimes called ’neural mass models’, have a long tradition (264; 552; 553; 16; 4; 180; 519; 20; 78; 163; 79; 183; 367; 371). Their aim is to predict the temporal evolution of the population activity $A(t)$ in large and homogeneous populations of spiking neurons.

Why do we restrict ourselves to large populations? If we repeatedly conduct the same experiment on a population of, say, one hundred potentially noisy neurons, the observed activity $A(t)$ defined in Eq. (12.1) will vary from one trial to the next. Therefore we cannot expect a population theory to predict the activity measurements in a single trial. Rather all population activity equations that we discuss in this chapter predict the expected activity. For a large and homogeneous network, the observable activity is very close to the expected activity. For the sake of notational simplicity, we do not distinguish the observed activity from its expectation value and denote in the following the expected activity by $A(t)$.

Why do we focus on homogeneous populations? Intuitively, we cannot expect to predict the activity of a population where each neuron receives a different input and has a different, and potentially unknown, set of parameters. However, if all neurons in a population have roughly the same parameters and receive roughly the same input, all neurons are more or less exchangeable and there is a realistic chance that, based on the knowledge of the parameters of a typical neuron in the population, we would be able to predict the activity of the population as a whole. The notion of a homogeneous network will be clarified in the following subsection. In Section 12.2.2 we will ask whether the requirements of homogeneous populations can be relaxed so as to include some degree of heterogeneity within a population.

# 12.2.1 Homogeneous networks

We study a large and homogeneous population of neurons. By homogeneous we mean that (i) all neurons $1\leq i\leq N$ are identical; (ii) all neurons receive the same external input $I_{i}^{\rm ext}(t)=I^{\rm ext}(t)$; and (iii) the interaction strength $w_{ij}$ for the connection between any pair $j,i$ of pre- and postsynaptic neurons is ’statistically uniform’. The notion will be made precise in Section 12.3, but for the moment we can think of connections inside the population of being either absent or ’roughly the same’, $w_{ij}\approx w_{0}$, where $w_{0}$ is a parameter. For $w_{0}=0$ all neurons are independent; a value $w_{0}>0$ ($w_{0}<0$) implies excitatory (inhibitory) coupling. Not all neurons need to be coupled with each other; connections can, for example, be chosen randomly (see Section 12.3 further below).

Example: Homogeneous population of integrate-and-fire neurons

In the case of leaky integrate-and-fire neurons, encountered in Chapter 1 and 5 (Section 5.1), the dynamics are

 $\tau_{m}{{\text{d}}\over{\text{d}}t}u_{i}=-u_{i}+R\,I_{i}(t)\quad{\rm for~{}}u% _{i}<\vartheta$ (12.2)

combined with a reset condition: if $u_{i}\geq\vartheta$ then integration restarts at $u_{r}$. A homogeneous network implies that all neurons have the same input resistance $R$, the same membrane time constant $\tau_{m}$, as well as identical thresholds $\vartheta$ and reset values $u_{r}$. Note that we have shifted the voltage scale such that the resting potential is $u_{\rm rest}=0$, which is only possible if all neurons also have the same resting potential.

We assume that a neuron is coupled to all others as well as to itself with coupling strength $w_{ij}=w_{0}$. The input current $I_{i}$ in Eq. (12.2) takes care of both the external drive and synaptic coupling

 $I_{i}=\sum_{j=1}^{N}\sum_{f}w_{ij}\alpha(t-t_{j}^{(f)})+I^{\rm ext}(t)\,.$ (12.3)

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 Eq. (12.3) 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 $w_{ij}=w_{0}$ and use the definition of the population activity, Eq. (12.1). We find a total input current,

 $I(t)=w_{0}\,N\int_{0}^{\infty}\alpha(s)\,A(t-s)\,{\text{d}}s+I^{\rm ext}(t)\,,$ (12.4)

which is independent of the neuronal index $i$. Thus, the input current at time $t$ depends on the past population activity and is the same for all neurons.

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.

# 12.2.2 Heterogeneous networks

Our definition of a homogeneous network relied on three conditions: (i) identical parameters for all neurons; (ii) identical external input to all neurons; (iii) statistically homogeneous connectivity within the network. We may wonder whether all three of these are required or whether we can relax our conditions to a certain degree (523; 97). Potential connectivity schemes will be explored in Section 12.3. Here we focus on the first two conditions.

Let us suppose that all $N$ neurons in the population receive the same input $I$, considered to be constant for the moment, but parameters vary slightly between one neuron and the next. Because of the difference in parameters, the stationary firing rate $\nu_{i}=g_{\theta_{i}}(I)$ of neuron $i$ is different from that of another neuron $j$. The index $\theta_{i}$ refers of the set of parameters of neuron $i$. The mean firing rate averaged across the population is $\langle\nu\rangle=\sum_{i}\nu_{i}/N$.

Under the condition that, firstly, the firing rate is a smooth function of the parameters and, secondly, that the differences between parameters of one neuron and the next are small, we can linearize the function $g$ around the mean parameter value $\bar{\theta}$ and find for the mean firing rate averaged across the population

 $\langle\nu\rangle=g_{\bar{\theta}}(I)$ (12.5)

where we neglected terms $(d^{2}g_{\theta}/d\theta^{2})\,(\theta_{i}-\bar{\theta})^{2}$ as well as all higher-order terms. Eq. 12.5 can be phrased as saying that the mean firing rate across a heterogeneous network is equal to the firing rate of the ‘typical’ neuron in the network, i.e., the one with the mean parameters.

However, strong heterogeneity may cause effects that are not well described by the above averaging procedure. For example, suppose that the network contains two subgroup of neurons, each of size $N/2$, one with parameters $\theta_{1}$ and the other with parameters $\theta_{2}$. Suppose that the gain function takes a fixed value $\nu=\nu_{0}$ whenever the parameters are smaller than $\hat{\theta}$ and has some arbitrary dependence $\nu=g_{\theta}(I)>\nu_{0}$ for $\theta>\hat{\theta}$. If $\bar{\theta}=(\theta_{1}+\theta_{2})/2<\hat{\theta}$, then Eq. 12.5 would predict a mean firing rate of $\nu_{0}$ averaged across the population which is not correct. The problem can be solved if we split the population into two populations, one containing all neurons with parameters $\theta_{1}$ and the other with parameters $\theta_{2}$. In other words, a strongly heterogeneous population should be split until (nearly) homogeneous groups remain.

The same argument also applies to a population of $N$ neurons with identical parameters, but different inputs $I_{i}$. If the differences in the input are small and neuronal output is continuous function of the input, then we can hope to treat the slight differences in input by a perturbation theory around the homogeneous network, i.e., a generalization of the Taylor expansion used in the previous paragraph. If the differences in the input are large, e.g., if only a third of the group is strongly stimulated, then the best approach is to split the population into two smaller ones. The first group contains all those that are stimulated and the second group all the other ones.