State-space kinetic Ising model reveals task-dependent entropy flow in sparsely active nonequilibrium neuronal dynamics

TL;DR

A state-space kinetic Ising model is developed to track nonequilibrium entropy flow in neural populations, revealing task-dependent dissipation in the mouse visual cortex and underscore the model’s utility in uncovering intricate asymmetric causal dynamics in neurons and linking them to behavior through the thermodynamic underpinnings of neural computation.

Abstract

Neuronal ensemble activity, including coordinated and oscillatory patterns, exhibits hallmarks of nonequilibrium systems with time-asymmetric trajectories to maintain their organization. However, assessing time asymmetry from neuronal spiking activity remains challenging. The kinetic Ising model provides a framework for studying the causal, nonequilibrium dynamics in spiking recurrent neural networks. Recent theoretical advances in this model have enabled time-asymmetry estimation from large-scale steady-state data. Yet, neuronal activity often exhibits time-varying firing rates and coupling strengths, violating the steady-state assumption. To overcome this limitation, we developed a state-space kinetic Ising model that accounts for nonstationary and nonequilibrium properties of neural systems. This approach incorporates a mean-field method for estimating time-varying entropy flow, a key measure for maintaining the system's organization through dissipation. Applying this method to mouse visual cortex data revealed greater variability in causal couplings during task engagement despite reduced neuronal activity with increased sparsity. Moreover, higher-performing mice exhibited increased coupling-related entropy flow per spike during task engagement, suggesting more efficient computation in the higher-performing mice. These findings underscore the model's utility in uncovering intricate asymmetric causal dynamics in neuronal ensembles and linking them to behavior through the thermodynamic underpinnings of neural computation.

Figures (18)

• Figure 1: Application of the state-space kinetic Ising model to two simulated neurons.A A schematic of the time-dependent kinetic Ising model for two neurons with field and coupling parameters. The links between the nodes represent the neurons' causal interactions with arrows indicating the time evolution from the past to the present. B Raster plots for the two neurons. The vertical axis represents the number of trials, and the horizontal axis shows the number of time bins. C The approximate marginal log-likelihood as a function of the iteration steps of the EM algorithm. D The optimized hyperparameter $\mathbf{Q}^{i}$ for neuron 1 (left) and neuron 2 (right). E (top) Estimated and true time-dependent field parameters. The solid lines represent the MAP estimates of the field (first-order) parameters obtained from the smoothing posterior, $\boldsymbol{\theta}_{t|T}$. The shaded areas show the 95$\%$ credible intervals derived from the diagonal elements of the smoothed covariance matrix, $\mathbf{W}_{t|T}$. The dotted lines are the field parameters from true $\boldsymbol{\theta}_{t}$ used to generate the data. (middle, bottom) Estimated and true time-dependent coupling (second-order) parameters.
• Figure 2: The application of the state-space kinetic Ising model to 12 simulated neurons.A Simulated spike data for the first, 100th, and last trial out of 200 trials. The vertical axis shows the number of neurons, and the horizontal axis represents the number of bins. B Estimated coupling parameters $\boldsymbol{\theta}_{t|T}$ (solid lines), for all neurons and time bins ($i=1,2,\ldots,12$, $t=1,\ldots,T$). Shaded areas indicate 95% credible intervals, and dashed lines denote the true parameter values used to generate the data. These plots show only the couplings that are significantly deviated from zero: The couplings whose 95% credible interval contains $0$ in all bins were excluded. For clarity, only five such significant incoming couplings from other neurons are shown in each panel. C Scatter plots comparing the true coupling parameters ${\boldsymbol{\theta}}_{t}$ with the estimated values $\boldsymbol{\theta}_{t|T}$ at time $t=10, 20, \ldots, 60$. The black line is a diagonal line.
• Figure 3: Estimation error and computational time.A Root mean squared error (RMSE) of the field and coupling parameter estimation as a function of trials $L$, with the number of neurons fixed at $N=80$. Results are averaged over 10 independent samples, with error bars representing standard deviations. B RMSE of the field and coupling parameters as a function of the number of neurons $N$, with the number of trials fixed at $L=550$. Averages and standard deviations are computed over 10 independent samples. C Average computation time for different numbers of neurons $N$ and trials $L = 55, 100, 300, 550$, with error bars indicating standard deviations. Computation was performed on a Dell PowerEdge R750 server with two Intel Xeon 2.4 GHz CPUs (76 cores / 152 threads).
• Figure 4: Comparison of entropy flow estimation methods. Entropy flows estimated using four different approaches: Sampling method with true parameters ${\boldsymbol{\theta}}_{t}$ (solid black); sampling method with estimated parameters ${\boldsymbol{\theta}}_{t|T}$ (dashed green); mean-field method with true parameters ${\boldsymbol{\theta}}_{t}$ (dashed blue); and mean-field method with estimated parameters $\boldsymbol{\theta}_{t|T}$ (solid red).
• Figure 5: Analysis of the model misspecification.A The population spike count histogram of $N=30$ neurons following the shifted-geometric model with a sparseness parameter $f=20$ and $\tau=0.8$ (empirical distribution obtained by the Gibbs sampling in blue circle; theoretical probabilities in green). The yellow line represents a distribution obtained from the state-space kinetic Ising model fitted to the Gibbs sampling data. B The activation function of the shifted-geometric model with $f=20$ and $\tau=0.8$ (green) and that of the kinetic Ising model (yellow) using the average of the fitted field and coupling parameters.