Figure 1

Relative firing rate $f$ and input ratio $\eta$ as a function of time, semilogarithmic scale. Each time step is $4\mathrm{ms}$. A total of 50 million time steps are taken for each simulation but only the first 20 million time steps are shown. These curves show that firing rate homeostasis by itself ($k_{11}$ and $k_{21}>0$, $k_{12}=k_{22}=0$) cannot guarantee critical homeostasis. With $k_{11}=2\times {10}^{-5}$ and $k_{21}=0.01$, the average relative firing rate converges to $f=1$ but the average input ratio converges to $\eta =9.8$. With $k_{11}=0.01$ and $k_{21}=2\times {10}^{-5}$, $f$ converges to one but $\eta$ converges to 57.2. With $k_{11}=0.01$ and $k_{21}=0.01$, $f$ converges to one but $\eta$ converges to $6.5\times {10}^{-4}$. All values are averaged over the prior one hour of data and over all $N=64$ nodes. The Hebbian learning factor in these calculations is $C_H=0.01$.Reuse & Permissions

Figure 2

Relative firing rate $f$ and input ratio $\eta$ as a function of time, log-log scale. Each time step is $4\mathrm{ms}$, $C_H=0.01$. These two examples show that firing rate homeostasis scaling of connection strengths ($k_{21}=0.01$, $k_{22}=0$) and critical homeostasis scaling of spontaneous firing probabilities ($k_{12}=2\times {10}^{-5}$, $k_{11}=0$) result in an unstable system. In example (a), the initial $S(i;0)$’s are too small to maintain firing rates and input ratios at unity. When the relative firing rates eventually drop below unity, the connection strengths $P(i,j;t)$ are scaled up and the input ratios rise. As the input ratios rise, the spontaneous firing probabilities are scaled down further, which causes the relative firing rates to drop even more, perpetuating a cycle such that the system eventually becomes silent $(f\approx 0)$ despite maximal input ratios [$\eta \left(i\right)\approx N$, where $N$ is the total number of nodes]. Varying the relative magnitudes of $k_{12}$ and $k_{21}$ does not improve stability. In example (b), the initial $S(i;0)$’s are too large, and the opposite situation arises, where a cycle is entered such that the spontaneous firing probabilities rise as the input ratios drop. This system then enters tonic hyperactivity with minimal connectivity.Reuse & Permissions

Figure 3

Relative firing rate $f$ and input ratio $\eta$ as a function of time, linear scale. Time step is $4\mathrm{ms}$. These examples show that the convergence criterion $\det \left(\mathbf{K}\right)>0$ does not guarantee convergence to the target asymptotic firing rate and input ratio. With $k_{11}=2\times {10}^{-5}$, $k_{12}=0$, and $k_{21}=k_{22}=C_H=0.01$, both $f$ and $\eta$ converge to unity. However, with $k_{11}=0.01$, $k_{12}=0$, and $k_{21}=k_{22}=C_H=2\times {10}^{-5}$, $f$ converges to 1.16 while $\eta$ converges to 0.84, even through $\det \left(\mathbf{K}\right)$ is identical in the two cases. All values are averaged over the prior one hour and over all $N=64$ nodes.Reuse & Permissions

Figure 4

Dependence on $k_D$. Example with $k_{11}=2\times {10}^{-5}$, $k_{12}=0$, $k_{21}=k_{22}=C_H=0.01$, time step of $4\mathrm{ms}$ and total of 50 million time steps. (a) Log-log plot of the distribution of avalanche sizes $G_A\left(n\right)$ vs avalanche size $n$. The solid line is a power law with an exponent of $-1.5$ shown for reference. (b) Semilogarithmic plot of relative connectivity vs distance, calculated by adding all $P(i,j;t)$’s at each distance and normalizing area under the curve to unity. Distances are in units of the lattice constant $a$, rounded to the nearest integral value. (c) Log-log plot of averaged spontaneous firing probability, input ratio, and relative firing rate as a function of $k_D$. (d) Semilogarithmic plot of distribution of connection strengths $P(i,j;t)$ vs connection strength, with total area under the curve normalized to unity.Reuse & Permissions

Figure 5

The spontaneous firing probability $S$, relative firing rate $f$, and input ratio $\eta$ as a function of time. A simulated seizure occurs between time steps $1.25\times {10}^8$ and $1.5\times {10}^8$ during which the activation levels $A(i;t)$ of 12 nodes were set to 0.8. In all examples, $k_{12}=0$, $k_{21}=k_{22}=C_H=0.01$, and the time step is $4\mathrm{ms}$. (a) The spontaneous firing probability averaged over all nodes $S$ declines with seizure onset and slowly recovers after the seizure ends. The rate of recovery is slower for smaller values of $k_{11}$. (b) The relative firing rate $f$ is elevated during the seizure and becomes depressed for a time after a seizure ends, more prominently so for smaller values of $k_{11}$. Note the semilogarithmic scale. (c) The average input ratio $\eta$ is depressed during a seizure and becomes elevated (i.e., the system is supercritical) for a time after a seizure. For $k_{11}={10}^{-6}$ and $k_{11}={10}^{-7}$, the input ratio remains supercritical for about 50 million time steps $\left(55\mathrm{h}\right)$.Reuse & Permissions

Figure 6

The spontaneous firing probability $S$, relative firing rate $f$, and input ratio $\eta$ as a function of time. Simulated acute deafferentation occurs at time step $1\times {10}^8$, at which time a system of 100 nodes is suddenly reduced to 10 nodes. Parameters are $k_{11}={10}^{-6}$, $k_{12}=0$, $k_{21}=k_{22}=C_H=0.01$, and the time step is $4\mathrm{ms}$. (a) The spontaneous firing probability $S$ as a function of time rises to a new steady state after acute deafferentation. (b) The relative firing rate $f$ declines with acute deafferentation and does not recover to steady-state baseline until $S$ has reached its new baseline. The input ratio $\eta$ compensates for the depressed relative firing rate by rising to supercritical levels and remaining there until $S$ and $f$ have both reached their new steady-state levels. The supercritical state in this example lasts for 8 million time steps $\left(9\mathrm{h}\right)$.Reuse & Permissions

Figure 7

Non-Markovian weighting function $G_0\left(t\right)$ as a function of time. In this example, $G_0\left(t\right)$ consists of a step function at short times plus a Gaussian centered at longer times. The areas under the step function and under the Gaussian are approximately equal.Reuse & Permissions