Homeostatic Adaptation of Optimal Population Codes under Metabolic Stress

Abstract

Information processing in neural populations is inherently constrained by metabolic resource limits and noise properties. Recent data, for example, shows that neurons in mouse visual cortex can 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. These dynamics are not described by existing mathematical models of optimal neural codes. We have developed a theoretical population coding framework that captures this behavior using two 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 that our method uniquely captures how populations of tuning curves adapt while maintaining homeostasis, as has been observed empirically.

Model

The optimization objective is defined as:

\[ \begin{aligned} &\operatorname*{arg\,max}_{g(\cdot), d(\cdot)} \int p(s)\, f\!\left(\eta_{\kappa}(E)^{-1} g(s) d(s)^2\right)\, ds \quad \text{(Coding objective)}, \\ &~~~~~\text{s.t.} \quad p(s) g(s) = R(s) d(s) \quad \text{(Approximate homeostasis constraint)} , \\ &~~~~~~~~~~\int p(s)\, g(s)^\alpha\, ds = E \quad \text{(Energy budget constraint)}, \end{aligned} \] where $p(s)$ is the stimulus distribution, $f(\cdot)$ is the coding objective (e.g. infomax or discrimax), $\eta_{\kappa}(E)$ is the energy-dependent noise level, $g(s)$ is the gain function, $d(s)$ is the density function, $R(s)$ is the firing rate, and $E$ is the energy budget.

Analytical Solution

	infomax	discrimax	$L_p$ error, $p = -2\beta$
Optimized function	$f(x)=\log x$	$f(x)=-x^{-1}$	$f(x)=-x^{\beta}, \ \beta < \frac{\alpha}{3}$
Density $d(s)$	$E^{\frac{1}{\alpha}} R(s)^{-1} p(s)$	$\propto E^{\frac{1}{\alpha}} R(s)^{\frac{-\alpha-1}{\alpha+3}} p(s)^{\frac{\alpha+1}{\alpha+3}}$	$\propto E^{\frac{1}{\alpha}} R(s)^{\frac{\alpha-\beta}{3\beta-\alpha}} p(s)^{\frac{\beta-\alpha}{3\beta-\alpha}}$
Gain $g(s)$	$E^{\frac{1}{\alpha}}$	$\propto E^{\frac{1}{\alpha}} R(s)^{\frac{2}{\alpha+3}} p(s)^{\frac{-2}{\alpha+3}}$	$\propto E^{\frac{1}{\alpha}} R(s)^{\frac{2\beta}{3\beta-\alpha}} p(s)^{\frac{-2\beta}{3\beta-\alpha}}$
Fisher info $FI(s)$	$\propto \frac{E^{\frac{3}{\alpha}}p(s)^2}{\eta_{\kappa}(E) R(s)^2}$	$\propto \frac{E^{\frac{3}{\alpha}} p(s)^{\frac{2\alpha}{\alpha+3}}}{\eta_{\kappa}(E) R(s)^{\frac{2\alpha}{\alpha+3}}}$	$\propto \frac{E^{\frac{3}{\alpha}} p(s)^{\frac{-2\alpha}{3\beta-\alpha}}}{\eta_{\kappa}(E) R(s)^{\frac{2\alpha}{\alpha-3\beta}}}$
Disc. bound	$\propto E^{\frac{-3}{2\alpha}}p(s)^{-1}$	$\propto E^{\frac{-3}{2\alpha}}p(s)^{\frac{-\alpha}{\alpha+3}}$	$\propto E^{\frac{-3}{2\alpha}}p(s)^{\frac{\alpha}{3\beta-\alpha}}$

Results

Comparison figure across conditions — Figure: 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 [Padamsey et al., 2022]). Our model (b, purple) predicts this flattening, while existing models predict either shortening (c, blue [Ganguli et al., 2010]) or widening (d, red [Wang et al., 2016]). (e) We compute the change in mean firing rate for each method under a uniform prior. Only our model exhibits firing rate homeostasis.

	infomax	discrimax	\(L_p\) error, \(p = -2\beta\)
Optimized function	\(f(x)=\log x\)	\(f(x)=-x^{-1}\)	\(f(x)=-x^{\beta}, \ \beta < \frac{\alpha}{3}\)
Density \(d(s)\)	\(E^{\frac{1}{\alpha}} R(s)^{-1} p(s)\)	\(\propto E^{\frac{1}{\alpha}} R(s)^{\frac{-\alpha-1}{\alpha+3}} p(s)^{\frac{\alpha+1}{\alpha+3}}\)	\(\propto E^{\frac{1}{\alpha}} R(s)^{\frac{\alpha-\beta}{3\beta-\alpha}} p(s)^{\frac{\beta-\alpha}{3\beta-\alpha}}\)
Gain \(g(s)\)	\(E^{\frac{1}{\alpha}}\)	\(\propto E^{\frac{1}{\alpha}} R(s)^{\frac{2}{\alpha+3}} p(s)^{\frac{-2}{\alpha+3}}\)	\(\propto E^{\frac{1}{\alpha}} R(s)^{\frac{2\beta}{3\beta-\alpha}} p(s)^{\frac{-2\beta}{3\beta-\alpha}}\)
Fisher info \(FI(s)\)	\(\propto \frac{E^{\frac{3}{\alpha}}p(s)^2}{\eta_{\kappa}(E) R(s)^2}\)	\(\propto \frac{E^{\frac{3}{\alpha}} p(s)^{\frac{2\alpha}{\alpha+3}}}{\eta_{\kappa}(E) R(s)^{\frac{2\alpha}{\alpha+3}}}\)	\(\propto \frac{E^{\frac{3}{\alpha}} p(s)^{\frac{-2\alpha}{3\beta-\alpha}}}{\eta_{\kappa}(E) R(s)^{\frac{2\alpha}{\alpha-3\beta}}}\)
Disc. bound	\(\propto E^{\frac{-3}{2\alpha}}p(s)^{-1}\)	\(\propto E^{\frac{-3}{2\alpha}}p(s)^{\frac{-\alpha}{\alpha+3}}\)	\(\propto E^{\frac{-3}{2\alpha}}p(s)^{\frac{\alpha}{3\beta-\alpha}}\)