Homeostatic Adaptation of Optimal Population Codes under Metabolic Stress

TL;DR

This work analytically derive the optimal coding strategy for neurons under varying energy budgets and coding goals, and shows how this method uniquely captures how populations of tuning curves adapt while maintaining homeostasis, as has been observed empirically.

Abstract

Information processing in neural populations is inherently constrained by metabolic resource limits and noise properties, with dynamics that are not accurately described by existing mathematical models. Recent data, for example, shows that neurons in mouse visual cortex go into a "low power mode" in which they maintain firing rate homeostasis while expending less energy. This adaptation leads to increased neuronal noise and tuning curve flattening in response to metabolic stress. We have developed a theoretical population coding framework that captures this behavior using two novel, surprisingly simple constraints: an approximation of firing rate homeostasis and an energy limit tied to noise levels via biophysical simulation. A key feature of our contribution is an energy budget model directly connecting adenosine triphosphate (ATP) use in cells to a fully explainable mathematical framework that generalizes existing optimal population codes. Specifically, our simulation provides an energy-dependent dispersed Poisson noise model, based on the assumption that the cell will follow an optimal decay path to produce the least-noisy spike rate that is possible at a given cellular energy budget. Each state along this optimal path is associated with properties (resting potential and leak conductance) which can be measured in electrophysiology experiments and have been shown to change under prolonged caloric deprivation. We analytically derive the optimal coding strategy for neurons under varying energy budgets and coding goals, and show how our method uniquely captures how populations of tuning curves adapt while maintaining homeostasis, as has been observed empirically.

Figures (14)

• Figure 1: A general solution. Our analytical framework generalizes previous work ganguli2010implicitwang2016efficient and predicts tuning curve flattening with a biophysical grounding in neuron simulation.
• Figure 2: Biophysical simulation of noise-optimal energy reduction. (a) We change cell properties of a simulated neuron to determine their impact on cell firing and energy use. (b) Single spike properties are extended to tuning curve properties under varying cell states. (c) We define optimal paths as those that minimize firing rate variance at each energy consumption level, predicting a specific noise/energy trade-off that we incorporate into our Fisher Information optimization framework.
• Figure 3: Simulation results of optimal adaptations under varying signal activity levels and intermediate outcomes. (a–c) Simulated results of total energy, and the mean and variance of spike count, respectively. (d) The energy corresponding to a fixed spike count, illustrated by the intersection in (a). (e) Fitted relationship between the mean and variance of spike count for an example pair of $v_{rest}$ and $g_{leak}$, used to define the dispersion $\eta_\kappa(v_{rest}, g_{leak})$. (f) Optimal adaptations obtained numerically by minimizing dispersion along each energy contour using the fitted energy and dispersion. (g) Optimal adaptations shown across different levels of signal activity $\kappa$. (h, i) The resulting density and dispersion as a function of optimal energy, respectively.
• Figure 4: Comparison of optimal tuning curves in control versus food-restricted mouse neocortex (L2/3). (a-d) The figure illustrates how an example tuning curve in an optimal population adapts to a tightening of energy-related constraints. Under metabolic stress, real tuning curves flatten (a, based on padamsey2022neocortex). Our model (b, purple) predicts this flattening, while existing models predict either shortening (c, blueganguli2010implicit) or widening (d, redwang2016efficient). (e) We compute the change in mean firing rate for each method under a uniform prior. Only our model exhibits firing rate homeostasis.
• Figure 5: Fitted parabolas for the mean and variance of spike count. We fit a parabolic relationship between the mean and variance of spike count across different cellular states $(v_{rest}, g_{leak})$. For visualization purposes, only every second value of $v_{rest}$ and $g_{leak}$ is shown.

Theorems & Definitions (4)

• Proposition C.1: Approximate Homeostasis
• proof
• Proposition E.1: Framework in ganguli2010implicit Implies Approx. Homeostasis
• proof