Dynamics of Competition between Subnetworks of Spiking Neuronal Networks in the Balanced State

Abstract

We explore and analyze the nonlinear switching dynamics of neuronal networks with non-homogeneous connectivity. The general significance of such transient dynamics for brain function is unclear; however, for instance decision-making processes in perception and cognition have been implicated with it. The network under study here is comprised of three subnetworks of either excitatory or inhibitory leaky integrate-and-fire neurons, of which two are of the same type. The synaptic weights are arranged to establish and maintain a balance between excitation and inhibition in case of a constant external drive. Each subnetwork is randomly connected, where all neurons belonging to a particular population have the same in-degree and the same out-degree. Neurons in different subnetworks are also randomly connected with the same probability; however, depending on the type of the pre-synaptic neuron, the synaptic weight is scaled by a factor. We observed that for a certain range of the "within" versus "between" connection weights (bifurcation parameter), the network activation spontaneously switches between the two sub-networks of the same type. This kind of dynamics has been termed "winnerless competition", which also has a random component here. In our model, this phenomenon is well described by a set of coupled stochastic differential equations of Lotka-Volterra type that imply a competition between the subnetworks. The associated mean-field model shows the same dynamical behavior as observed in simulations of large networks comprising thousands of spiking neurons. The deterministic phase portrait is characterized by two attractors and a saddle node, its stochastic component is essentially given by the multiplicative inherent noise of the system. We find that the dwell time distribution of the active states is exponential, indicating that the noise drives the system randomly from one attractor to the other. A similar model for a larger number of populations might suggest a general approach to study the dynamics of interacting populations of spiking networks.

Fig 1. Subnetworks and connections between them to form the full network.

The subnetworks are comprised of either excitatory or inhibitory neurons. Subnetworks B ₁ and B ₂ consist of neurons of the same type, either excitatory (the EEI scenario) or inhibitory (the EII scenario). The subnetwork A has neurons of the opposite identity. For excitatory (inhibitory) B ₁ and B ₂, we arrange w ≥ 1 (w ≤ 1).

Fig 2. In the J − w parameter space, three different types of dynamical behavior is observed for the EEI scenario.

(1) Equal rate (ER) for small values of w, in which the two excitatory subnetworks have identical non-zero firing rates. (2) For intermediate values of w, the two identical populations keep switching the activity between themselves, and none of them maintains the high activity, so called “winner-less competition” (WLC). (3) Large values of w result in winner-take-all (WTA) dynamics in which, depending on the initial conditions, one of the two competing populations maintains the high activity.

Fig 3. Switching dynamics of the two excitatory subnetworks for J = 0.1 mV and w = 2.5.

The population histograms of the three subnetworks with time intervals of 10 ms are plotted in light colors. A Savitzky-Golay filter with length 2n + 1 = 21, dt = 10 ms and polynomial order m = 4 was used to smoothen the signals, which are plotted in darker colors. The activity of the inhibitory subnetwork has less fluctuations compared to the activity of the excitatory subnetworks.

Fig 4. Joint distributions of population activities.

A. Interactions between total excitation and inhibition. The elliptical shape of the distribution indicates a strong positive correlation between excitation and inhibition. B. Negative interactions between the two excitatory populations. As activity is almost mutually exclusive, the term switching seems adequate. The unit of the numbers on the color bars are ms².

Fig 5. Inhibitory firing rate as a linear function of C and quadratic function of D.

A. Data from numerical simulations of the network are illustrated as colored points. The color bar shows the firing rate of the inhibitory population in unites of spikes per milliseconds (same unites for C and D). Black contour lines show the isolines of Eq (13) after parameter estimation. The equal distance between the iso-lines indicates a linear relationship between the inhibitory firing rate and C. Unites of the numbers on the color bar are kHz. B. Data points for a constant level of C = 0.9 kHz are illustrated in blue dots as a scatter plot of D and I. Model fit for all available data points after plugging C = 0.9 into Eq (13), shows the quadratic dependence of I on D (red curve).

Fig 6. Flow of the two excitatory subnetworks in two dimensions, extracted from simulated time series (top) and inferred from the Lotka-Volterra model (bottom).

A. State space of excitatory-excitatory firing rates extracted from simulations. B. Sum-difference state space extracted from simulations. C. State space of excitatory-excitatory firing rates are inferred from the model. D. Flow of the system in Sum-difference state space inferred from the model.

Fig 7. Flow of the system Eq (12) for w = 1.

Only one fixed point at D = 0 exists, and the sum of the activities of the excitatory populations is nonzero. The equilibrium is unique, therefore stationary activity without any switching results.

Fig 8. Life time statistics of the winning (high rate) population.

A. Difference between the rates at the stable fixed point as a function of the parameter w that describes the relative strength of synapses within each population. B. Survivor function of the distribution of life times for the two competing excitatory populations. C. Average life time as a function of w for network simulations. D. Difference of energy levels (see text for details) between the attractor and the saddle point of Eq (10), when the inhibitory population firing rate is replaced by the other two dynamical variables, as a function of w.

Fig 9. Variance of the noise in the state space of the two competing excitatory subnetworks.

A. Variance of ${\stackrel{.}{E}}_1$ and B. Variance of ${\stackrel{.}{E}}_2$ in the state space spanned by E ₁ and E ₂.

Fig 10. Excitatory population firing rate in the C − D coordinates, inferred from simulation results for a network in the EII scenario.

A. Excitatory firing rates corresponding to different values of C and D are shown in different colors (see the color bar). Parameter estimation for Eq (13) shows the estimated values of the excitatory firing rate. The isolines for the model are depicted in black. B. Excitatory firing rate for a constant value of C = 0.64 kHz are illustrated in blue dots. For this constant, Eq (13) predicts a quadratic scaling of the excitatory rate with the dynamical variable D. The red curve is the model prediction, with parameters mentioned in the text, for the constant value of C. The dependence of the excitatory firing rate on C and ∣D∣ are positive and negative, respectively.

Fig 11. Two-dimensional flow characterizing the firing rate dynamics for an EII network.

A,C. Excitatory-excitatory firing rate state space. B,D. Sum-difference state space.