The reason for employing the described scaling mechanism this time instead, is that this transformation of the degree spaces preserves the shape of F (κ) for the chosen ensemble of networks, if 1/s is included as coupling strength. Because the Heaviside step function provides for some variable x and constant c the property θ(cx) = θ(x), we can evaluate the system for different scaling factors always at the identical steady state solution, if we scale the threshold value appropriately, namelyΘ∗ = 1sΘ. Because then θ(F (κ∗)− Θ∗) =
θ(1sF (κ)− Θ∗) = θ(F (κ)− Θ), as we saw in Fig. 2.14.
Summarizing, we have seen that increasing the number of nodes per population by itself does not suffice to make the population activity model arbitrarily precise in the description of steady states of the system. On the contrary, for sufficiently large population numbers, the steady state profile converges to a sigmoid shape. This shape is explained by the fluctuations δn, which can be decreased by increasing the degree for all possible degrees and appropriate rescaling of the threshold value.
We have seen that the ensemble of networks with flat degree distribution is an ideal system to study the population activity model, because it allows the analytical solution for the functionsF (κ) and accordingly steady state conditions. Real world networks however show various degree distributions. In the following, we briefly demonstrate the applica- tion of the population activity model to an ensemble of networks with a different degree distribution, namely the Erdös-Rényi (ER) random graph.
2.4 Application to networks with other degree distributions
We study ER random graphs [35] of similar size and mean degree as the networks studied before. Specifically we choose the number of nodes to beN = 100, 000 and a mean degree of ⟨k⟩ = 200. For the network creation we used the Igraph software library for complex network research [28]. Fig. 2.15 shows the joint degree distributionN (k, k′), measured from the network realization (in blue), compared to the corresponding distribution calculated according to Eq. (2.3.2) for an uncorrelated random network with a binomial degree dis- tribution P (k) =(Nk−1)pk(1− p)N−1−k. Therein p denotes the probability that any two
nodes share a link. For better visibility, only every fourth degree is displayed in the plot. In this example the valuep = 0.001 was used both for the creation of the network and for the calculation of the degree distribution and hence joint distribution function.
The joint distribution function measured from an exemplary network realization agrees reasonably well for intermediate values ofk, while there are significant deviations towards the smallest and largest measured values ofk. This echoes bad statistics, which are caused by the very few actual nodes of these degrees. The binomial degree distribution of ER random graphs has technically no upper or lower bound for the degrees, as with a small probability very small and very large degrees are in principle possible. The shown limits in Fig. 2.15 for the calculated joint distribution function result from the smallest and largest observed degrees in the example network realization.
Fig. 2.16 shows a summary of the simulation results for the ER graph. Panels (a) and (b) show the total population activity Nkuk and the population activity uk at different time
2 Population equations for degree-heterogeneous neural networks
k
0 160 180 200 220 240 260 160k
180200 220240 260N(k
,k
0)
0 2 4 6 8 10 12 14Figure 2.15. Joint distribution function in the ER network. The calculated N (k, k′) from a real-
ization of a random graph as described in the text (in blue), compared to the joint distribution function calculated from Eq. (2.3.2) (in red). The degree distribution is given by a binomial distribution. The measured and calculated joint distribution functions agree reasonably well for intermediate values ofk andk′, but differ significantly towards the maximum and minimum degree k.
to calculated population activity fronts that were initialized asuk(t = 0) = θ(k− 185) and
uk(t = 0) = θ(k− 186), respectively. The former initialization leads to population activity
fronts that travel towards a stable steady state solution, while the later initial condition results in vanishing activity.
Fig. 2.16(c) shows stable steady state front positions ˜κ (gray marks) for a number of different threshold values Θ. The cyan marks show the smallest initial degrees that lead to vanishing activity in the network, hence they can be related to the regime of unstable steady state solutions. The initial conditions for the example of Θ = 158, cf. (a) and (b), are indicated by vertical dashed lines in corresponding colors.
The solid red line in Fig. 2.16(c) shows F (κ) calculated from the joint distribution function according to the theory, Eq. (2.2.15). The dashed red line showsF (κ) calculated from the measured joint distribution function, cf. Fig. 2.15. For a particular threshold Θ, the branch left of the zenith marks the onset of front positions that are expected to travel towards larger population degrees. The measured functionF (κ) agrees very well with the simulation results (cyan marks). Note that the measured and calculated functions F (κ) and G(κ) lie close to each other, for which reason both functions G(κ) are not shown.
The large deviations towards the lower and upper end of the degree distributions in Fig. 2.15 are echoed in the simulation results of the activity model as the theory allows the calculation of F (κ) essentially for every possible degree. Nevertheless, for the smallest degrees, the stable steady state front positions are not recovered by the simulation. In
2.4 Application to networks with other degree distributions 150 200 250
k
0 1000 2000 3000N
ku
k (a) 150 200 250k
0.0 0.5 1.0u
k (b) 150 175 200κ
140 145 150 155 160 165 170 175Θ
(c) F (κ) calculated F (κ) measuredFigure 2.16. Population activities in the ER random network. Snapshot of the total population activity for two different initial conditions (uk(t = 0) = θ(k− 185) and uk(t = 0) = θ(k− 186)) in
(a). The corresponding population activity in (b). The functionF (κ) calculated and measured from the network realization (solid and dashed line, respectively) is shown in (c). Gray marks show simulation results of stable steady state front positionsκ˜sfor different threshold valuesΘ. The example shown in
(a) and (b) is indicated by the dashed horizontal line. Gray and cyan vertical dashed lines show the two initial conditions in corresponding color to (a) and (b).
this particular example the correct stable steady state solution is calculated for population degrees aboveκ = 153. In the example realization of the network, the number of neurons of all populations including and below this degree was21 of 100000. The smallest observed degree was 146.
In summary, analyzing simulation results for the steady state population activity in Erdös-Rényi networks shows reasonable agreement with the population activity model. Stable and unstable step degrees are recovered for a wide range of parameters. This be- comes challenging towards the limits of the degree distribution, as these populations are occupied by very few neurons only. Nevertheless, this example demonstrates the applica- bility of population activity model to a class of networks that is arguably more relevant than networks with flat degree distributions.
2 Population equations for degree-heterogeneous neural networks