Feedback control stabilization of critical dynamics via resource transport on multilayer networks: How glia enable learning dynamics in the brain

Learning and memory are acquired through long-lasting changes in synapses. In the simplest models, such synaptic potentiation typically leads to runaway excitation, but in reality there must exist processes that robustly preserve overall stability of the neural system dynamics. How is this accomplished? Various approaches to this basic question have been considered. Here we propose a particularly compelling and natural mechanism for preserving stability of learning neural systems. This mechanism is based on the global processes by which metabolic resources are distributed to the neurons by glial cells. Specifically, we introduce and study a model composed of two interacting networks: a model neural network interconnected by synapses that undergo spike-timing-dependent plasticity; and a model glial network interconnected by gap junctions that diffusively transport metabolic resources among the glia and, ultimately, to neural synapses where they are consumed. Our main result is that the biophysical constraints imposed by diffusive transport of metabolic resources through the glial network can prevent runaway growth of synaptic strength, both during ongoing activity and during learning. Our findings suggest a previously unappreciated role for glial transport of metabolites in the feedback control stabilization of neural network dynamics during learning.

Figure 1

Glial-neuronal interactions: (a) Cartoon based on existing experiments, illustrating how glia serve to distribute metabolic resources from the bloodstream to neural synapses. Red arrows indicate paths of metabolite transport. (b) A simplified directed graph representation of our two-layer network model. Black arrows indicate neural synaptic interactions. Arrow thickness indicates synaptic strength which evolves according to STDP. Red arrows which terminate on black arrows represent the resource supply to the corresponding synapse.

Figure 2

Resource-transport dynamics stabilizes network activity (Experiment 1): (a) Time series of ${\lambda }^t$ (largest eignenvalue of $W^t$) reveal rapid convergence to stable network dynamics ($\lambda \approx 1$), independent of initial conditions. Three different initial conditions were tested: hyperexcitable (blue, ${\lambda }^0=1.5$), stable (black, ${\lambda }^0=1$), and hypoexcitable (red, ${\lambda }^0=0.5$). The inset is an expanded view of the first 5000 time steps. (b) After a longer transient the total resource $\mathcal{R}$ also stabilizes to a steady value. (c) Similarly, in all three cases, the average activity $S$ reaches a statistical steady state with large fluctuations.

Figure 3

Turning off diffusion results in runaway growth (Experiment 2): (a) The maximum eigenvalue $\lambda$ versus $t$, and (b) the total resource $\mathcal{R}$ versus $t$. The data plotted in black are “baseline” results obtained using our model as described in Sec. 2 of the paper. For the data plotted in red (labelled “instability”), the initial evolution is the same as for the baseline data up until $t=100000$ (marked in the figure by a vertical arrow), at which time the diffusion of resources between the glial cells is turned off, as described in the text.

Figure 4

The STDP network learns and remembers (Experiment 3): We divide the neurons into two equally sized groups, $G_1$ and $G_2$, consisting of 500 neurons each. This results in four groups of synapses: synapses within the first group (Within $G_1$), synapses that convey signals from neurons in $G_1$ to neurons in $G_2$ ($G_1$ to $G_2$), synapses that convey signals from neurons in $G_2$ to neurons in $G_1$ ($G_2$ to $G_1$) and synapses within the second group (Within $G_2$). Panel (a) depicts the learning protocol (see text). Panels (b) and (c) show $\lambda$ and $\mathcal{R}$ versus $t$. The learning regime spans $t=[80000,100000]$ (delimited by the vertical arrows). Panel (b) shows that $\lambda$ becomes subcritical during learning [35], but then quickly evolves back to the critical state $\lambda \cong 1$. Panel (d) shows the mean synaptic strength for the four groups of synapses for excitatory synapses during learning. In accord with the STDP learning rule, the mean synaptic strength increases for $G_1$ to $G_2$ synapses. In the post-learning regime, spanning $t=[80000,160000]$, panel (d) shows that the model remembers what it learned.

Figure 5

Robustness of dynamics to parameter change: Time average of the largest eigenvalue, ${\langle \lambda \rangle }_t$, as a function of $C_1/{\widehat{C}}_1$, $C_2/{\widehat{C}}_2$, and $D/\widehat{D}$ where ${\widehat{C}}_1$, ${\widehat{C}}_2$, and $\widehat{D}$ are the parameter values used for Figs. 2, 3, 4. All three curves show that our model is fairly robust to parameter changes, e.g., a 25% change in $C_1$ or $C_2$ yields a change in ${\langle \lambda \rangle }_t$ of about 0.3%.