Homeostatic Adaptation of Optimal Population Codes under Metabolic Stress
Authors: Yi-Chun Hung
Gregory W Schwartz
Emily A. Cooper
Emma Alexander
Venue: ICLR 2026 Poster (2026)
Keywords: Efficient neural codes
neural computation
metabolism
Primary area: applications to neuroscience & cognitive science

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.

Published as a conference paper at ICLR 2026
HOMEOSTATIC ADAPTATION OF OPTIMAL POPULATION CODES UNDER METABOLIC STRESS
Yi-Chun Hung', Gregory W. Schwartz', Emily A. Cooper Emma Alexander'
I DDepartment of Computer Science, Northwestern University
Departments of Ophthalmology and Neuroscience, Feinberg School of Medicine, Northwestern University Heerbert Wertheim School of Optometry & Vision Science, University of California, Berkeley
HHelen Wills Neuroscience Institute, University of California, Berkeley
ychung@u. northwestern edu emiilycopeerberkel weweeerreeeeeeeeeeeeee...e. weeeleey.eeee edu {greg. . schwartz, ealexander eeleeaanderrenootttwettttt.. hwwoot/wweteeeeeee...
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 adap- tation leads to increased neuronal noise and tuning curve flattening in response to metabolic stress. These dynamics are not described by existing mathemati- cal models of optimal neural codes. We have developed a theoretical population coding framework that captures this behavior using two surprisingly simple con- straints: an approximation of firing rate homeostasis and an energy limit tied to noise levels via biophysical simulation. key feature of our contribution is an en- engy budget model directly connecting adenosine triphosphate (ATP) in cells to a fully explainable mathematical framework that generalizes existing optimal population codes. Specifically, our simulation provides energy-dependent dis- persed Poisson noise model, based on the assumption that the cell will follow an optimal decay path produce the least-noisy spike rate that is possible given cellular energy budget. Each state along this optimal path is associated with prop- erties (resting potential and leak conductance) which can be measured in electro- physiology experiments and have been shown to change under prolonged caloric deprivation. We analytically derive the optimal coding strategy for neurons un- der varying energy budgets and coding goals, and show that our method uniquely captures how populations tuning curves adapt while maintaining homeostasis, has been observed empirically.
INTRODUCTION
Animals have limited access to calories, which constrains their brains' ability to expend energy to represent information. Additionally, energy budgets are inconsistent over the lifetime of an organ- ism, which must adapt in times of scarcity while maintaining as much functionality as possible. Padamsey al. have recently shown that under long-term calorie deprivation, mice lose the ability to discriminate fine-grained visual details while keeping coarser information, and that they do so with systematic changes in biophysical cell properties that result in flattening of neuronal tuning curves (Padamsey et al. (022). Perhaps surprisingly, this energy-saving strategy does not result in a significant change spike counts, demonstrating the importance of firing rate homeostasis even when energy budgets tight.
This energy-saving strategy motivates us to develop an analytical and simulation-groundee frame- work that describes how metabolicalll--ttessse neurons can optimally change to produce noisier, cheaper spike trains rather than reducing their firing rates, consistent with Padamsey et al.'s empir- ical observations. Firing rate homeostasis under metabolic stress, along with direct but tractable accounting for ATP use, is the key to correctly modeling how real tuning curves change. The stan- dard analytical approaches of limiting spike rates (Ganguli & Simoncelli 2010 Wang et al. 2016a)

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can predict optimal populations under single metabolic state, but fails to capture how codes adapt to changing conditions.
The efficient coding hypothesis (Barlow ct al. [961] asserts that populations of sensory neurons
optimized to maximize information about the environment subject to resource constraints. While
the impact of coding objectives (Ganguli & Simoncelli 2010 2014 Manning
al. 2024 Wang
et al. 2016a) and noise properties (Laughlin et al. 1998 Manning et al. 2024] Eoker et al.)
2011
Wang et al. 2016a) on optimal population codes have been studied in depth, the effect of energy constraints and in particular their variation over time remains underexplored. Simple models of
energy constraints (e.g.. on mean firing rate in Ganguli & Simoncelli (2010): Yerxa al. (2020);
Rast & Drugowitsch (2020) or maximum firing rate in Wang et al. (201 2:016al0 b): Laughhin (1981) are
analytically convenient, but they fail to capture the complex trade-offss determining energetic costs in real cells, such as subthreshold activity and biophysical adaptations (Padamsey & Rochefort 2023). We will show that these simplified constraints lead to inaccurate predictions of neurons' adaptation to metabolic stress (i.e., reduction of energy budget). Specifically, our model predicts tuning curve flattening closely in line with Padamsey et al. (2022), while previous population coding models capture either the shortening (Ganguli & Simoncelli 2010)) or widening (Wang et al. 2016a) behavior of energy- limited tuning curves, but not both.
We capture this behavior with two novel constraints: energy budget (linked directly to ATP con- sumption, firing rate noise, and easily-measured cell properties) and an approximation of homeosta- sis. Remarkably, our hoppysicalll-motivaae strategies; instead, they provide simple extension of existing models to characterize the dynam- ies of neural coding strategies under metabolic stress. The contributions of our framework are as follows:
constraints do not complicate the analysis of coding
We introduce two mathemaaticallyytractabb constraints that are the first to accurately de- scribe real neurons' recently-characcterized response to metabolic stress.
Our framework generalizes previous models, and recover disparate results from the literature with simplified parameters, offering a united mathematical explanation for the partial successes of previous work.
Biophysical simulation grounds key parameters of our model, including a dispersed Pois- son noise model that realistically describes signal degradation calorie-deprived neurons.
RELATED WORK
Optimal neural codes. Since the foundational work by Barlow et al. (1961), the efficient cod- ing hypothesis has been extensively evaluated across a variety of contexts, leading to the develop- ment of numerous information-theorrtie models with strong empirical support from both neural and perceptual data (Ganguli & Simoncelli 2016 Manning et al. 2024 Vacher & Mamassian [023]
Risau-Gusman 2020 Zhaoping et al.) 2011; Laughlin 1981). A variety of coding goals have been
analyzed, from entropy maximization Laughlin] 1981) to mutual information maximization (Brunel
& Nadal 1998 Wei & Stocker] 2016 Wang al.] 20. 6 Manning al. 2024 Ganguli & Simon-
celli 2010; to L, norm error minimization (Wang et | 20
Ganguli & Simoncelli 2210
2010
2014 Manning et al. 2024).
Optimal coding models are often built on Fisher information (FI) Waan et al. 2016a; Ganguli & Simoncelli 2010 2014 Manning al. 2024 Wang et al. 2016b), which describes the local sen- sitivity of a population response around stimulus value. One reason for this is that FI can be estimated more easily from measurements of tuning curves than entropy-based information met- rics. Another is that Fl optimization is closely related to standard coding objectives: maximizing log Fl approximates mutual information maximization (Brunel & Nadal 1998 Wei & Stocker, 2016), while maximizing power functions of FI minimizes Lp loss in the asymptotic limit (Wang) ct al. 2012). Coding goals of particular interest here are the "infomax" (mutual information max- imization, through log Fl maximization) and "discrimax" (discriminability maximization, through -FI- maximization, implying L2 error minimization by the Cramér Rao bound) objectives.
Optimal neuronal populations can be predicted directly from Fl-based objectives, provided that sig- nificant simplifying assumptions are imposed on tuning curve shapes. For example, Wei & Stocker 2015) assumes homogeneous tuning widths amplitudes across the population. Usefully, Ganguli
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compact representation of population codes through two continuous functions: gain and density, denoted by g(-) and d(-), re- spectively. These functions are then used to modulate base shape (e.g., Gaussian or sigmoid) to construct heterogeneous population codes. Wann et al. 2012) also adopt the tiling assumption, with fixed gain function.
& Simoncelli (2110) introduced tiling assumption that enables
Constrained optimization. Constraints shape optimal codes just as strongly as coding objectives. Constraints in prior literature can typically be categorized into three types: energy, coding capacity, and population size. Energy-basee -ased constraints, such as those based on mean firing rate (Ganguli & Simoncelli 2010 Yerxa et al. 2020, Rast & Drugowitsch, 2020) or maximum firing rate (Wang et al. 2016a Laughlin 1981), reflect the energetic costs of neuronal firing. Coding capacity, expressed as the sum of the square root of FI, has been employed in Wei & Stocker (2016); Wang al. (2012) and generalized to other power laws in Morais & Pillow (2018). The population size constraint Ganguli & Simoncelli (2010); Yerxa et al. (2020) reflects the finite number of neurons available during neural processing.
The joint influence of coding objectives and constraints on shaping optimal population codes has also been explored in recent work. MMrais & Pillow (2018) show how direct trade-offs between coding objectives and Fl-based constraints can obscure the relationship between these factors. |Rast & Drugowitsch (020) describe an adaptation procedure to jointly determine objectives and constraints from neural data, but their model does not allow changes in co-firing patterns like those caused by tuning curve widening. Most similar to our work, Wang et al. (2016b) analyzes ON-OFF population codes under metabolic stress by modeling both energy costs and noise levels power laws of firing rate. However, their energy cost model is inconsistent with Padamsey et al. (2022) and they do not connect the two terms through noise-energy trade-off. None of these constraints connect directly to ATP use, due to the biophysical complexity of real neurons.
Energy costs beyond spike counts and homeostatic mechanisms. ATP the energy currency of cells, yet most models of energy consumption in neurons use the simplifying assumption that spikes have a fixed cost and dominate the neuron's ATP usage. In fact, spikes are not typically the largest part of neuron's s ATP budget, costing only 22% of energy use (Harris al., 2012). Reversal of ion fluxes from postsynaptic receptors (50%) and reversal of leak sodium entry via the Na*/K* exchanger (20%) together dominate ATP usage in neurons (Harris et al. al. (012)). We use biophysical model of neuron that estimates ATP usage from all three of these sources, creating a more realistic energy constraint than spike rate alone.
In fact, one of the key features of our neuron simulation is that it estimates different energy rates while maintaining fixed spike rate. This result is consistent with Padamsey et al. (2022) and relies on systematic changes in cell parameters observed in metabolically stressed mouse cortex. Homeostatic maintenance of fixed firing rate commonly observed feature in neural circuits, often involving the balance of excitation and inhibition (Liang et al. 2024).
It worth noting that the approximating a neuronal energy cost through firing rate commonly adopted in recent literature [Tavoni 2025; Koren & Panzeri 2022 Koren et al. 2025 Gutierrez Denéve 2019) for analyzing changes in intrinsic cell-level and/or network level properties. Al- though these works use firing rate as an energy proxy same in Ganguli & Simoncelli (2010); Yerxa et al.] (2020); Raat & Drugowitsch] (2020); Wang et al. 2016aab); Laughlin (1981) they do not consider the effect of metabolic stress on neurons, reported in Padamsey et al. (2022). In fact,
these models cannot reiterate a central finding of this recent work: that spike rates remain the same energy use is reduced.
Reeeoely-disooveredd neural mechanism. Padamssy et al. (2022) show that neurons change their biophysical state so that they can maintain their mean firing rates while expending less energy. Primarily, they modify electrical properties to reduce the current associated with receiving input spikes, because reversing this current costs significant ATP through ion pumps. Essentially, by sitting closer to the spiking threshold and receiving smaller signal from each incoming spike, the cell uses less energy for signal-carrying spikes but also becomes readier to fire in response to stochastic inputs and system noise, which makes them more vulnerable to false firings. Because these false firings are strictly positive, they raise the zero values at the edges of their tuning curves (widening), which requires compensatory drop in peak values (flattening) to maintain homeostasis.
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3 ANALYTICAL MODEL
We consider a population of neurons jointly encoding a scalar stimulus with their mean firing rate determined by tuning curves (s)) Assuming conditionally independent firing rates, the Fl the population can be computed from the tuning curves as
(s)
Fi(;;
(1)
Var(hn E) E))
==1
Based on the neuron simulation described in Sec. 4 energy-varying noise can be expressed as a dispersed Poisson distribution, with dispersion factor (E') that captures variability in neural responses under different activity levels & and energy budgets E:
War(hn(s); ;); E) =M (E))..s))
An optimal population, under energy budget E, maximizes the expectation with respect to proba- bility p(s) of target function f on the FI, while maintaining firing rate homeostasis (expected firing rate Rn)) within each neuron:
p(s)s (FT(s s.t. ((()))n()) Run Vn.
(.)
argmax ha hex())-.
(3)
Notably, [(x = log x, the formulation corresponds to infomax; whereas if f(x) = -x-1, corresponds to discrimax.
3.1 APPROXIMATION WITH TILING
Solving eq. (3) is challenging without parametrizing the tuning curves (s), due to the vast space of arbitrary continuous functions. To mitigate this difficulty, parameterize the tuning curves using two continuous functions - gain g(s) and density f(s)-followin the approach of Ganguli & Simoncelli (2010). Specifically, the tuning curve is defined as h (s) g(s)n(D(s) D(s,)), where 8n denotes the preferred stimulus of the n-th neuron. Here, D(s) represents the cumulative function of the density d(ss) given by D(s) S d(s) ds. We assume that p(s) varies more smoothly than any tuning curve (s) and analyze the case in which the base shape i((-) is Gaussian.
To approximate the summation in eq. 3 in terms of (s)) and d(s), we adopt the same tiling prop- Wang et al. 2012). Under this assumption, the Fisher
erty as in Ganguli & Simoncelli 2010 information is approximated as:
Fl(s; E) ~ nhe (B) (())))))) gs))d(s)²
(4)
We can approximate the homeostasis constraint, and generalize it to the population level, as follows:
p())g(s) R(s)d(s).
(5)
This approximate homeostasis constraint be visualized as rectangular (height (ss),, width 1 1/d(s,) approximation of each Gaussian tuning curve and a locally constant p(s). Detailed deriva- tions of eqs. 4 5 provided in Secs. A to
Combining the approximations eq. and eq. 5 gives the following population optimization:
argmax [ p(ss)f (B) ( (()))(s))))) ds, s.t. p(s)g(s) = R(s)d(s)-
(6)
g(-),d())
is evident that this optimization problem is only bounded when an additional constraint is imposed on g(s) and/or d(s)) We propose the following generalized energy constraint:
f p(s)g(s) E, where 1.
This model is consistent with Padamsey al. (2022), which shows that energy savings come from reducing synaptic conductance (analogous to gain) rather than other cell parameters that widen the tuning curve (analogous to density), which we interpret as compensation maintain homeostasis. We will relate the energy budget to physical energy expenditure and fix the value of a for real neurons in Sec.
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3.2 ANALYTICAL SOLUTION
We analytically solve the proposed optimization framework for various objective functions (see Tab. 1 by eliminating d(s) via the approximate homeostasis constraint and applying the Lagrangian method. Here, we outline the key steps in the derivation for the infomax case, with other objective functions and full details included Sec. D To solve the optimization problem defined by eqs. ( and (7). we eliminate d(s) by substituting from the approximate homcostasis constraint in eq.
yielding the following formulation:
/ p(s) log
ds, s.t. J ((())(()) ds E.
max g(-)
(8)
The corresponding Lagrangian with multiplier À is given by:
((g. = f p(s) log (()
ds ~
Setting 0,C 0 and using eq. 7 provides the optimal gain, with density following from eq. (5)
g(s) =E
(10)
d(s) -/ ((s)R[(s)
(11)
The resulting is obtained with eq. (4) and the discriminability bound from the Cramér Rao bound, summarized in Tab. along with the analytical solutions corresponding to I.p error objective func-
tions. interesting to point out that our model predicts that both density (i.e., 1 1/width) and gain
scale on order of E1/a, while the Fisher information scales on the order of 7/ (E) g3/a, Additionally, show in Sec. N that the analytical solutions in eqs. (TO) and (11) are robust to differential correlations, which are the dominant violation the independent noise assumption in real neurons [Moreno Bote al. (204)). This robustness extends to significant changes in noise correlations across metabolic states.
Table 1: Analytical solutions in our framework. For comparison to other methods, see Sec. L, error, p
infomax
discrimax
-23
Optimized function
f(x)
log
f(x)
f(x) x² 8
Density d(s)
E R(3) p())
a
E R(s) p(s)a13
x EE [(>
Gain g(s)
Ed
E ((a)(s)
E= R(s) R=
Fisher info F1(+)
F p(p(o)
α Eap(s)
Eap(
nn RE)
Disc. bound
E p(s)
E (
3.3 RELATION TO EXISTING MODELS
The approximate homeostasis and energy constraints introduced in eqs. 55) and 7 are not only grounded in biophysiological evidence but also generalize commonly used constraints in existing models (see Tab. 2). Here, we first demonstrate how our constraints relate to those in Ganguli & Simoncelli (2100). followed by a discussion the connection to Wang et al. (2012). It is evident that energy constraint reduces to the mean firing rate (FR) constraint used in Ganguli & Simoncelli (010)) when setting a = 1 and redefining as the mean firing rate denoted as R. The population size constraint can also be recovered from constraints follows:
E f p(s)g(s) ds approx. homeo.
/ R(s)d(s)
R(+)== E N d(s))ds+ fdss) N.
(12)
By jointly enforeing the conditions that yield the mean FR and population size constraints, obtain R(s) = R/N, indicating that the population mean firing rate R is evenly distributed across neurons.
It is important to note that Ganguli & Simoncelli (2010) uses the parameter R to denote the total expected firing rate of a neuron population. The parameter R, in eq. (3) represents the expected firing rate of the n th neuron in the population. Meanwhile, R(s) can be interpreted as the expected
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firing rate of a neuron whose preferred stimulus is and can be regarded "continuous" version of Rs
The derivation in eq. 12) leads to observe that approximate homeostasis, though not explicit requirement of Ganguli & Simoncelli (2010), emerges as a signature of optimality in their model (see Sec. E for proof of this proposition).
To reduce our constraint to the constraint of total FI states in Wang et al. (012). we now set a = 3/2:
/ p(s)g(s)² ds upprox homen.
/ R(s) (d(s)g(ss²
ds
VFI
(13)
R())= E / / (((s))g(sdds
/ (ss))
where IR the FI coding capacity. Under the assumption a = 3/2, the above reduction implies that energy constraint be interpreted as weighted version of coding resource constraint, where the weighting function corresponds to the firing rate R(s). The generalization of our model to the existing frameworks is summarized and illustrated in Fig.
In summary, we show that our general case can be reduced to previous models the literature. Specifically, by setting
and using firing rate that of Ganguli & Simoncelli (2010). Furthermore, when a = 3/2 and the Fl coding capacity is incorporated, our energy constraint reduces the coding capacity constraint proposed in Wang al. (2012). In the following section, we will describe the biophysical parameters that reduce the general case to a true ATP-based energy constraint.
energy proxy reduces our constraint to
Comparison of different works on neuron population codes. Key parameters include the total number of neurons (N), mean firing rate (R, as constant or function of stimulus s), Fisher information coding capacity (C), energy budget (E).
Table 2:
Works
Constraints
Constraints under Tiling
Noise
Low Energy
Canguli & Simoncelli (2010)
FR population size
J (s))g(s)ds R. d(s)ds N ((3) xI
R 4. shortens
Poisson
(N fixed)
max FR
Poisson or
C 4, widens
Wang et al. (2016a)
total FI states
gg))d(() C
Gaussian
fixed)
Ours
energy
(s)g(s)dds g Ric)
Energy-dependent
flattens
approx. homeostasis
Dispersed Poisson
(H(s) fixed)
d(a) p(a)
Our Framework
Energy Constraint
Constraint with Homeostasis
Reduced
Framework
R R(s)
Rial
Ganguli & Simoncelli
/ ((()gg(s)ds= E
d(e)
population size
=
(shortens)
g(s) α R(s) =
ufa) d(e)
R(a)
Wang al. (widens)
Ours, general
I ((s))(s) =5
total states
Constant firing rate
a
Cell Simulation
Ours, specific (flattens)
lowest noise available
E-0a
ATP use
Figure 1: A general solution. Our analytical framework generalizes, previous work (Ganguli & Si- moncelll 2010, Wang et al.) 2016a) and predicts tuning curve flattening with a biophysical ground- ing in neuron simulation.

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4 BIOPHYSICAL SIMULATION
We employ the biophysical simulator NEURON (Hines et al. [002 to investigate the trade-offs be- tween noise and energy. We simulate a single-compartment neuron based on a stochastic Hodgkin- Huxley model in response to an input spike, under a variety settings. Specifically, we vary the three cellular parameters identified by Paaaamsyy et al. (2022) changing under metabolic stress: resting potential Urest, leak channel conductance Gleak. and synaptic conductance Jsyn- See Fig. P for an overview of the simulation pipeline, Sec. Sec.[foo for an intuitive explanation of each parameter's effect as well as full list of simulation parameters, Sec. G for details of energy accounting, and the supplement for full simulation code.
Value ranges for Urest and gleak follow those of Padamsey et al. (022). and we vary Jeyn from O to 250 pu/cmm?. This range allows us to exclude cell parameter triplets that lead to mean spike count over .8 or under .2 total during the 2 second simulation duration in the deterministic setting, so that stochastic simulation captures meaningful variance. From 10, 000 simulated trials for each condition, evaluate the mean and variance of the spike count, denoted ja and of. as well as the energy expenditures associated with maintaining the resting state and signal activity (combining the cost of receiving and generating spikes), denoted as Ebg and Esig. respectively.
We then derive higher level statistics from the single-spike characteristics. First, we compute the energy cost ttoaal associated with each cell state. This cost combines rest state and signaling energy, weighted by an activity level к (see Sec. H.T This factor is key for modeling visual neurons, which have orders-of ooders-of-maaniuude changes in spike rates along the visual pathway (Goris al., 2014). Then, we use the variance of spike count to characterize noise dispersion, details in the following section. Finally, we follow Padaassyy et al. (2022) to measure tuning curve widening w by asserting Gaussian tuning curve via rescaled synaptic conductance Jeyn. details in Sec. H.2 Note that this procedure associates a unique value of 9ayn to each (Ureat, 9teak) pair, reducing the degrees of freedom in cell parameter space.
4.1 NOIRE- OPTIMALITY
Using the noise of and energy Etotal associated with each cell state (Urests gleak: k),, we assert two hypotheses in turn to predict biophysical and information theoretic changes under metabolic stress, see Fig. 3 First, we assert strict homeostasis constraint to extract cell parameters corresponding to constant mean spike count (white plane in Fig. B1) and extract the corresponding energy costs (Fig. 3p. energy costs across the extracted plane shown in d). We note in Fig. B that firing count variance saturates and then drops as Joyn increases, as the probability of firing approaches 1. The variance is quadratic in mean spike count, consistent with Bernoulli distribution. but scaled by a factor that grows under increased metabolic stress. Specifically, we fit the scale of the firing variance parabola for the dispersion 7)::
of = = The Mr - (1-)).
(14)
Fig. 3e illustrates this fitting for a single (Urest: Meak) state. Although we fit the T/e based on mean and variance of spike count under single spikes, it can be generalized to spike trains (see Sec.
Blophysies Simulator
Energy Terms
Energy Cast
(c) Optimal Noise EneseyEnergy Trade-offs
Cell Properties
Single Spike Properties
Ebg
Nolse Energy Modeling
Optimal Adaptation
Esig
Clotal
KCai Cg
Ureat
Spike
Spike
Awag
9leak
25 ms
Ebg caip
2 F
of
FR Model
1]
écotul
Copt
Resented
To Tuning Curve
glook
9eym
Conductance
Width
Carrent
Menbrenc Potential
Oaym
Tuning Curve Model
W
Vreal
Figure 2: Biophysical simulation of noise-optimal energy reduction. (a) We change cell proper- ties 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 spe- cific noise/energy trade-off that we incorporate into our Fisher Information optimization framework.
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We then assert second assumption: that the cell will adapt so that for every energy budget, noise will be minimized. Fig. 3 shows a heatmap of 77 associated with each cell state, with white lines indicating level sets of energy cost. Along each constant-energy line, the lowest noise point lies on the orange curve, which describes an optimal decay path through the cell parameter space. Specifically, this path is the solution of the following constrained optimization problem:
Sapt (E; k) = argmin The (Urest: Nleak)
s.t.
Etotal Etotat(Urest (Urests Peeaki k) =.
(15)
Veest; Sloak
To numerically solve this minimization problem, we 77x using a 17 17-parameter model and intersect it with linear fit of Etotal (see Sec. H.4 for fitting details). Each "trest-9leaks cell state along the optimal path Saet (c, x) corresponds to both an energy cost and noise level, providing the empirical trade-off between these terms, as well as a tuning curve width. Note that the shape of this curve varies with activity level K, as shown in Fig. Fig. As Frest becomes less negative, signaling becomes cheaper but maintaining resting potential is more expensive, so only active cells deviate signifi- cantly from -75 mV. Additionally, changing the balance background vs. signaling energy terms changes the slope of constant-enen constant-energy energy-contonn to generate paths different curvature (see Sec. D.
which intersects with the nonlinear noise landscape
This simulation fixes the unknown terms and 7), from our mathematical model as follows. First, note that our optimal densities in Tab. are all proportional to E
(a) Mean of Spike Count PP
(b)
Total Energy Stotal
Variance of Spike Count
0.1
0.1
120
120
120
Boak JAS cm 2
120
JAS] fem
70
70
Reest [mV]
Usest [mV]
Ureat [mV]]
(e5) cm
-66
Senn p15)
66
10/ A [a)
Brw juS/om
466
0.4
3.15
0.0
0.05
0.1
0.15
0.20
0.25
(d)
Grotal
(e)
of
n (Dhun Abaa) up (1 1a vi)
Fit
100
1.056
3.7
Etata! t1o" ATP/))
Occu | eem"
110
0.2
Stook JuS] cm
110
1.045
3.3
80
femm [is
1.034
2.9
90
90
1.023
60
Buu
66 mVV
Kuils.
1.012
Upeat (77 V
-74
[20 (TT/TP
2.1
0.0
9hak T0pSS
0.4
-40
70
Ureat
[mV]
-74
1.001
(g)
Ne (Mreet Staa))
(h)
(1)
90
2.3
L.05
1.05
Do 0.99
Sloak JuS (em=
150
2,5
2A4
L.045
1.034
0.94
0.96
1.03
1.04
0.93
90
0.92
90
L012
0.90
120
70
1.001
0.88
1.00
2.1
2.9
3.7
4.5
Ureat [m]]]
Exgt [10 ATP//
Eigh ATO'ATP//s
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 fixed spike count, illustrated by the intersec- tion in (a). (c) Fitted relationship between the mean and variance of spike count for an example pair of Urest and fleak, used to define the dispersion 7)s (Erest, Jleak). (f) Optimal adaptations obtained numerically by minimizing dispersion along each energy contour using the fitted energy and disper- sion. (g) Optimal adaptations shown across different levels of signal activity k. (h, i) The resulting density and dispersion as a function of optimal energy, respectively.
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When we define E in units of ATP/s
E(K) 01 (k)) + a2(k),
(16)
get the optimal density scales d(s) E (ai (K)eopt + (x))) Since Fig. 3 shows that, across activity levels, the optimal density has an affine relationship with Copt. it follows immediately that
1.
(17)
See Sec. J for fits of ā(k: values from simulation, as well as B(kc), ((e), and (u())) defined below.
The noise dispersion along the optimal adaptations shown in Fig. follows the intuitive trend that decrease in energy leads to an increase in noise. Higher activity and tighter energy constraints lead to larger changes in noise levels. In general, the noise dispersion takes the form
(K(K) The (Copt) (mk) + Copt (2()
ei (K)
(x())) + E(x) ca(n)
(18)
showing direct trade-off between noise and energy in both physical units and the analytical energy budget E. Returning to our optimized solutions in Tab. [ we note that noise only indirectly affects the optimal density and gain, by shaping the optimal path and therefore setting a. also directly scales the Fisher information by a factor of 7h (E) allowing empirically-derive 1/2 values to function as a plug-and- and-play term when estimating Fisher information for subsequent analyses, such as the evaluation of perceptual thresholds.
5 CONSISTENCY WITH DATA
Here, we will show that our method accurately predicts the tuning curve flattening effect seen in mouse cortex (Padamsey et al., 2022), while existing models incorrectly predict shortening or widening. We assume infomax-optine ulus, and explore predicted changes under metabolic stress (see Sec. L for details). Mouse data in Padamsey et al. (2022), illustrated in Fig. 4h. shows 32% widening of the tuning curve and statistically insignificant change in firing rate, which we model as an exact match in spike count. In Fig. 4p. our model reiterates the 32% widening and 29% reduction in ATP consumption reported in Padamsey et al.] (2022). To accomplish both simultancously. we fit and 02 from eq. (16) to 0.19625, where car denotes the energy expenditure the control condition. In the infomax case with uniform prior, we achieve a perfect tuning curve match because our approximation homeostasis is exact. See Sec. [ for homeostatic errors other settings.
Gaussian tuning curve for mniformly-distribun
stim-
satisfy (2/((])ctr)
We compare our method to existing, non-homeostatic frameworks (Ganguli & Simoncelli, 2010 Wang et al. (016a). In Fig. Gffguii & Simoncelli (2100) constrains the mean firing rate directly, with fixed tuming curve width derived from the (fixed) population size. Because the tuning curve cannot match the widening observed in data, we reduce the firing rate to match the tuning curve peak instead. This requires direct 24% reduction in the firing rate, violating homeostasis by -24% under our uniform prior assumption. Fig. 44, Wang et al. (2016a) maintains a constant max firing rate while reducing coding capacity, with 24% reduction in coding capacity required to widen by 32%, resulting in a homeostatic violation of 32%. See Sec. M for discussion of model likelihoods.
6 DISCUSSION
Changes in energy availability are a fact of life for all sensing organisms, and a key factor in evolu- tionary fitness. Indeed, there is good evidence that neural circuits maintain spike rate homeostasis across range of contexts, perhaps
essential part of keeping activity in neural circuits bal- anced (Hengen et al. 2013 Wu et al. 2020)..
We have characterized the noise-energy trade-off in a simulation that connects tractable and gen- eralizing mathematical framework directly to ATP use. also makes specific, testable predictions about how basic biophysical properties of cells should change under metabolic stress, factoring in activity level changes across the visual pathway. Specifically, the optimal paths in Fig. B3g predict specific relationships between resting potential and leak conductance for neurons of different spike

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Published as a conference paper at ICLR 2026
(a) Mouse Neuron
(b)
Ours
(c)
Ganguli & Simoncelli
(4)
Wang al.
(c)
30
32%
Cuntrol Food-restricted
20-
Menn FR
Flatten
Shorten
Widen
FX
10-
0%
0
-10
-20
-301
-24%
Stimulus
Figure 4: Comparison of optimal tuning curves in control versus food-restricted mouse tex (L2/3). (a-d) The figure illustrates how an example tuning curve in an optimal population adapts 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 mod- els predict either shortening (c, blue Ganguli Simoncelli (2010)) or widening (d, red Wang et al. (2016a)). (e) We compute the change in mean firing rate for each method under umiform prior. Only our model exhibits firing rate homeostasis.
rates they face metabolic pressure. These properties readily measured in patch-clamp exper- iments, and future work will perform these experiments in matched types of neurons from control and food-restricted animals.
In summary, we analytically derive and predict that both density (i.e., 1/width) and gain scale on the order of Ei/a, while the Fisher information scales on the order of (E)) /3/aa (see Sec. 3). Addi- tionally, demonstrate that model encompasses the competing models (Ganguli & Simoncelli 2010 Wang et al.) 2016a) as special cases. As a result, it preserves their explanatory power and. in particular, matches the ability of Ganguli & Simoncelli (2016) to produce signatures of optimality that have been found in brain areas for five measured attributes (three visual and two auditory). In Sec. we use stochastic Hodgkin-Huulle-llnn
model calibrate the parameters, yielding a and The (E)= 1/0 + E-cz" Finally, in Sec. 5 we show that the analytical solution with the calibrated and T) accurately predicts the tuning-curve flattening effect reported in Padamsey et al. (022)..
Future work can address current limits of our model. Mathematically, we seek to provide more general tractable approximations of homeostasis conditions, particularly for non-Gaussian and non- tiling populations. Sigmoids and Gabors are tuning curve families of particular interest. Our single- compartment biophysical neuron model can also be extended capture higher-fidelity cell behaviors.
more complicated simulation to also possible that cell recordings will show that, contrary to our central assumption, metabolic stress responses are driven primarily by non-coding factors and are not information information-thhcortccalll theoretically optimal, calling for additional modelling.
Tractable, grounded models of metabolic impacts neural codes are needed in order ask new kinds of optimization questions, interpret existing data, and incorporate accelerating progress in our understanding of metabolism. The new optimization questions can ask include: how popula- tion codes be optimized to handle a range of energy budgets? Does an animal's 's lifetime or early life access to calories impact their neural codes? What animal from how their codes change under metabolic stress? In short, we can consider optimality in terms of long-term survival strategies rather than at snapshot states. The answer these questions may lead to reinterpretations of existing data that has been used as evidence of particular optimiza- tion functions, as in Manning et al. 2024); Ganguli Simoncelli (2016). And while/Padamser 2022) describes changes that occur on the timescale of weeks, it is possible that some metabolic stress responses occur more quickly, with implications for the interpretation of neural data from food-deprived animals, common in behavioral literature. Though model is based heavily data from Padamsey et al. (2022), rapidly emerging understanding of the importance of metabolic and homeostatic factors in visual system function and nervous system health (Walls et al.) 2025 Lin
tell about the needs and priorities of an
al.)
et al. 2025 Etchegaray & Ravichandran 2011 Meng et al.] 2025) Kim et al. 2025
al.
2024 suggests that more data in this area may be avai available
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ACKNOWLEDGMENTS
This work was supported by GaiwanNoortwwesten Doctoral Scholarship. We would like to acknowl- edge Mark Agrios and the NSF-Simons National Institute for Theory and Mathematics in Biology (NITMB) for valuable discussions.
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A APPROXIMATE FISHER INFORMATION
We assume following
Convolution tiling property: Σn
(D()) Loonv h(D(s)-D(ea)
g(s) is much smoother than the function h(s).
Each neuron responds independently.
The first assumption is commonly adopted in the previous literature (Ganguli & Simoncelli, 2010 Wang et al.) 6a) and the second one is implicitly used in Ganguli Simoncelli(0010) Based on these assumptions,
approximate the total Fisher information FT(s) as:
(s)²
FI(8; E)
Σ
(If dispersed Poisson and by independence)
The (s)
- D(ss))d(ss
(E)
g(s)h(D(s) - ((nn))
(19)
= =n) - g(s)d(s)P (sn)))
h(D(s) D(s.))
~ g(s))))))
The approximation relies the assumptions. In our analysis, we further assume that the constant Teomy is equal to 1, as it does not influence the analytical solution of the proposed framework. For the cof-cconiitenncy, we also derive the first equality Sec.
B DISPERSED POISSON NOISE MODELING
Assuming that the firing rate of the n-th neuron, denoted r.. is sampled from dispersed Poisson distribution approximated by ((ha(s) (s), ne hE),,, (s)), we can derive the Fisher information (FI) as
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follows:
(
( d [llog((nnss) 2
E); = B
(20)
Os
-1
r (s())
(r- hn (
-N),()))
(21)
(s)
The ())
2 hz (s)2
3
1
-(G,()))²
(22)
(s)
nhe (E)(n))))))
(sn
(s)
(s)
(s)
(23)
2hn(s)
(s)2
(24)
The (E),,..,
where the second line follows from the definition FI, and the third line is obtained by applying mo- ment properties of the Gaussian distribution. The last approximation is based on the non-negligible
firing rates (e.g., (s) >> Be(E)) ). Note that the second term in eq. 23 is due to the Gaussian approximation of the dispersed Poisson. It can more elegantly addressed by introducing the quasi-likelihood function McCullagh (2019).
C APPROXIMATE HOMEOSTASIS
Proposition C.1 (Approximate Homeostasis). Assume the following:
The probability density p(s) and the gain g(s) of the stimulus s are much smoother than h(D(ss) - D(sx)), where D(s) is the cumulative function of the density d(s).
The functions hi(-) and ((s) are differentiable.
Then, the following approximation holds:
I p(s)h) (())))))((((S) - D(s.)) ds p(sn)
g(sn) d(sn)
f h(x)) dr.
(25)
In the proof, we first exploit the smoothness of the probability density p(s), followed by first- order Taylor approximation of D(s) - Ds*.) around 8,, and finally apply the change of variable (s sn).
Proof.
J (s)) p(s)g(s)h (D(s) D(sn)) ds
((on)g(on) / (D(ss) D(sn)) ds
(26)
~ x((n)g((n)
(d(sn))s sn)) ds
= =((s)) g(Sn)
/
(((z)d dx,
d(sn))
where the first approximation uses the smoothness assumption of p(s), and the second approxima- tion applies first-order Taylor expansion of D(s) around sn-
Followed by the approximation in theorem C.I we further relax the approximation to the population level and assume the normalized h((x)) Then, we obtain the approximate homeostasis constraint:
p(s) ((s) = R(s).
(27)
d(s)
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D DERIVATION OF ANALYTICAL SOLUTIONS
D.1 INFOMAX
/ p(s) p(s) log (())))))()) ds
argmax (())
s.t. p(s) g(s)
R(s))
(28)
d(s)
/ p(s)g(s)(ds= E, where a 2 1.
By eliminating d, we have
argmax f p(s) log ((())))(())(p(()) a(-)
ds, s.t. (D(e)a(o)dds= E.
(29)
The corresponding Lagrangian with multiplier À is given by:
((g, = p(s) log (n, (F) (())) ds ((0))(())ds ds r.). (30)
Setting 0yL = 0, we obtain
ac 3p(s)
Xap(s)g(s)a-1 =0
(31)
0g g(8)
3
g(s) Aa
(Assume p(s) 0).
(32)
Then using the energy constraint, we have
/ p(s)
3
p(s)g(s)'ds
ds
(33)
da
λa
Therefore, get the optimal solution:
g(s) E//0 d(s) E1/0 pps).
(34)
(35)
D.2 DISCRIMAX
/ -p(s) ((())))))))) -1 ds
argmax .().(())
p(s) g(s) d(s) R((s)
(36)
/ p(s)g(s) = E, where 2 1.
By eliminating d, we have
/ -p(s) 9())
ds, s.t. f [(s))()))ds= E.
argmax a(-)
(37)
The corresponding Lagrangian with multiplier À is given by:
((g,A) f -p(s) (nw (F)g()) pp(s)> ?(())?) ds -2 (ʃ (1o)o(o0-d- r²).
(38)
Setting 0,C= 0, we obtain
OL ())))) \ap(s)g(s)0-1== =0
(39)
0g
g(8) (3)((E))(s))
+t
(40)
)ap(s)2
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Then using the energy constraint, we have
E / p(s)g(s)'ds=
/D(S)
(3nn(E)
ds
(41)
{xxp()) ())
(3m)(F)) (F)) da
at3
203 (fD()) (((s) afd R(s) ds
(42)
Adize
A 3n (E)
Adisc E
(43)
α
where we denote the integral in eq. once the integral has been performed, but will vary if p(s), R(s), changes. Because this term will not change the shape of any predicted distributions (c.g.. gain, density, or Fisher information), suppress these dependencies in our notation convenience.
(42)
as a scalar Adisc- Note that Adise is constant with respect to
Plugging À to g()), we have
(3 (3)((E))(s))
37/ (T))E))) (()
((5)
(44)
{xp((82
3n.(E)
Adias
ap(s)2
R(s)2
(45)
( p(3)2
F R(s) a+3
x E= R(s)
(46)
diecP(s)
Therefore, we get the optimal solution:
g(s) x E R(s) (
(47)
d(s) x E R(s) Jं #1
(48)
D.3 GENERAL
/ -p(s) (ne(E)) (()))))))))) ds
argmax s().d(.)
s.t. p(s) g(s) d(s)
R(s)
(49)
/ ((8)g(s) = E, where a 2 1.
By eliminating d, we have
argmaa / (p(ss) a(-)
ds, stt. d(sssss) E.
(50)
The corresponding Lagrangian with multiplier À is given by:
C(g. = / pps)) (E) R(s) 20) ds > ( (D(s))(s)= E). (51)
Setting 0,C 0, we obtain
Oc p(s) -3,0m (F (0()) 1(()2 ( 28 )=)()))(s)0~11= =0, (52)
0g
The Tha(E)) (F) R(s)33 It'a
Aae
g(s)
(53)
33
23
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Then using the energy constraint, we have
E f p(())(s))ds
33 nh (b)))
Xa
30-a
p(s))
55-0
R(())
38-0 ds.
(54)
Agen
33 A The (E)))
-8
(55)
A gen
Therefore, get the optimal solution:
g(s) x E R(s) ((())
(56)
d(s) x E R(s) -
(57)
E RELATION BETWEEN APPROXIMATE HOMEOSTASIS CONSTRAINT AND OPTIMALITY
Proposition E.1 (Framework in Ganguli & Simoncelli (2010) Implies Approx. Homeostasis). Let the optimizaatin problem be formulated as
[ 1())) (g(()d(s))) ds,
max g(),d(.)
S.I.
p(s)g(s)ds= R. f. [(s)ds= N,
(58)
where and N are constants. Then any optimal solution (g",d") must satisfy
g(s) P(s)
R
(59)
d(s)
N
Proof. We form the Lagrangian:
L(g, d, 21, J. p(s){(o(s)d(()))
> (ʃ p(o))(o)dd
(60)
d (falo)de N
Taking the first-order conditions, we have:
0L (f(x)
p(s)
d(s)² >ii(()) 0,
(61)
0g dr
0L
2p(s) ff(z)
dr g(8)d(s) A2 = O.
(62)
0d
Dividing the two equations:
Aa
8f(a) 2p(
(())d(8)) 2p(s)g(s)
Ox
(63)
X1
0f(z)
p( 8 ds))
d(s)
dr
Since Å1 >a constants (w.r.t 8), the ratio 2p(s)g(s) must also be constant. Letting C get
d(s)
2X,
p(s) g(8)
C.
(64)
d(s)
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To identify this constant, multiply both sides by d(s) and integrate:
J. ( c f. d(s)ds,
(65)
= R
(66)
Thus, the optimal solution must satisfy
p(s) g(s) R d(s) N
(67)
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F SIMULATION OVERVIEW
Recall that current reversal is a significant energy cost, so that at a high level each of these parameters has the following effect:
Resting potential Crest: When Urest is less negative, receiving spikes is cheaper. because it takes less of a voltage change (and therefore less current) to reach the spiking threshold. Input current must be reversed by ion pumps using ATP. Perhaps counterintuitively, a less-negative trest also makes sitting at rest more expensive. This is because a less negative Ureat shifts the balance between the reversal potentials of Nat and K+, which requires counteracting higher influx of Na and costs more energy to maintain (see Sec. G for details and relevant equations). This cost trade-off between maintaining rest potential and reversing the current associated with incoming spikes is why the optimal paths shown in Fig. 3 vary in shape as a function of the input activity level of the cell.
Leak channel conductance gleakt Leak channels are major factor in the overall membrane conductance. Using V = IR I/g we see that this mechanism also reduces the total current needed make a voltage change in the cell.
Synaptic conductance Jeyn: This prevents the cell from firing excessively by counteracting the changes in Ureat and gleak that reduce the energy cost of reaching spiking threshold. It scales down the impact of each incoming spike so that the mean firing rate of the cell is maintained. Essentially, for a cell that is twice as ready to fire, we adjust the synaptic conductance so that each incoming spike only creates half the change it usually would. This rescaled state is not identical in function to the original because the smaller distance to threshold, particularly in combination with the higher voltage variability also observed in this state, results more accidental output spikes. (We will illustrate this effect with a figure of simulated voltage traces, showing cell that requires three sequential input spikes to fire: the well-fed case, large voltage steps will cause an output spike even in the presence of noise, but in the starving case, two smaller steps result in smaller gap from threshold, which can be overcome by noise and cause an unintended output spike.) Note that this variable is how we inject specific tuning curve into the model, following Padamsey et al. (0022).
We additionally summarize how we configure the parameters in the simulation (see Tab. 3) and list the notation (see Tab. 4h in this paper.
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Table 3: Simulation Parameters
Category
Value
membrane capacitance membrane leaky channel conductance (control)
1F/emm
0.12 mS/ m
membrane reversal potential for passive current (control)
-75mV
membrane action potential threshold
-50mV
ion channel model file
stch carter subtchan.ses
Na* reversal potential
55 mV
Kt reversal potential
-90 mV
Na" channel total conductance
35 mS/cm"
single Na* channel conductance
20 pS
Na channel stochasticity
False
Kt Delayed Rectifier total conductance
mS/ InS/em²
single K* Delayed Rectifier conductance
20 pS
K Delayed Rectifier stochasticity
False
K' subchan total conductance
0.18 mS/cm²
single K* subchan conductance
20 pS
K subchan stochasticity
True
soma diameter
8 jmm
soma length
8 jmm
synapse input type
synaptic conductance ms
synapse rise time constant (double-exponentic model) synapse decay time constant (double-exponential model) synapse reversal potential synapse input noise model stimulus spiking event stimulus stochasticity simulation initial potential simulation duration simulation step
ms
0 mV
((gayn
Payn
10
single spike
False
-75 mV
2000 ms
0.1 ms
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Table 4: Notation in modeling and framework
Name
Notation
resting potential
Urest
membrane leaky channel conductance
Gteak
Maximum synapse conductance
9enn
mean of spike count
pr
variance of spike count
oF
background energy consumption
Cbg
signal energy consumption
Esig
mean of firing rate
PFR
variance of firing rate
Himullusddpendde synaptic conductance (gaussian shape) tuning curve width
G(s)
signal activity total energy cost
R
Etotal
optimal adaptation
S (c)
noise dispersion along optimal adaptation
T (E)
stimulus
$
preferred stimulus of the n-th neuron
8n
total Fisher information
FI(s)
Fisher information of the n--h neuron
FL.(s)
objective function
f())
stimulus probability density function gain
p(s)
g(s)
density
d(s)
base shape (gaussian) tuning curve of the n-th neuron
h(s)
hn(ss
firing rate of the n-th neuron
Rn
firing rate of approximate homeostasis energy cost the energy constraint total number of neurons
R(s)
N
exponent in the energy constraint exponent in the objective function of the general case
a
3
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G BIOPHYSICAL SIMULATION
The overview of the simulation pipeline is shown in Fig. 2 We focus three key cellular properties - resting potential Prest, leak channel conductance fteak and synaptic conductance Jxy as the primary variables in our simulation, based on findings by Padamsey al. (2022), who reported that neurons adapt their encoding approach through changes in these parameters.
We employ a single-compprttment simulation model that includes three ionic channels, one passive (leaky) channel, and a double-exponential conductance-based synapse (see Tab. 33. Additionally, the ionic channel modeling files are based on Padamsey et al. (2022). A single input spiking event is delivered to the synapse at the beginning of each simulation trial. In the simulation, we configure a dinee-iinnnional parameter space defined by Urest, Pleak- and Jsyn- Specifically, Urest and Gleak varied over the ranges of 75 -5 mV and 0.07 to 0.12 mS/cnts respectively. For each pair of Urest and Gleak. we further tune 9emm to ensure that the resulting mean spike count remains within physiologically valid range based on the Jayn values (corresponding respectively) obtained from the deterministic settings. It should be noted that we slightly abuse the notation Jsyn to refer specifically to the maximum synaptic conductance synaptic model. At each sampled point in the parameter space, we run the simulation 10, 000 times for estimating the mean and variance of the spike count. We should note that we simplify the change of Urest as the membrane reversal potential for passive current due to the large conductance of the leaky channel compared to others. To estimate the energy cost during active and silent signal periods, integrate the temporal current traces associated with the synapse, the Na" ion channel, and the leaky Na" ion channel. The integral of the synaptic current trace is identified as the signal-related energy expenditure, denoted Esig. while the sum of the integrals of the Na* current from the ion channel and Na" leak currents is defined as the background energy expenditure, Ebg- It is important note that the majority of the synaptic current is concentrated within the first 100 ms, owing the 10 ms decay constant used in modeling the synaptic conductance. Additionally, we also convert the integrated currents into ATP by assuming that one ATP molecule is required to pump out three Na" ions, consistent with the Na" Ma*/K*-AN AAPPase pump [Thoaa 1972). Since our simulation only adopts single leaky channel summarizing all the leaky currents, we use the following formula for estimating the Na* leak currents:
μF 0.2 and μæ 0.8,
the double-exponenttal
Na (t)
Na (Usoma (t)
Na
(68)
where Usoma (t) and E are the simulated soma temporal potential and Na* reversal potential, respectively. We further estimate the conductance of the leaky Na* ion channel, denoted as Reak by
Na Deeak
gteak/ +r),
(69)
= EENa"
Eleck
Eleak
EK
(70)
where Eleak and EK' denote the leaky channel and K* reversal potentials.
The above simulation was performed on the distributed cluster computer based on Intel(R)
Xeon(R) and does not require any GPU usage. We distributed the simulation of each cell states (Urest, Steak, Steak) to a core with individual 1G memory. Each simulation that executes 10,000 run for estimating the mean and variance of spike count takes approximately 40 minutes.
H BIOPHYSIOLOGICAL MODELING
H.1 ENERGY COST
We estimate the total energy cost as:
(71) where represents the level of signal activity. In our simulations, is varied over the range from 90 to 150. A detailed analysis of how different values of x influence the noise term The along the optimal adaptation path is provided in Sec.
étotal = Keeig
Ebg:
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H.2 TUNING CURVE WIDTH
Following Padamsey et al. (2022). we determine the Joyn that yields μa 0.1 for each state
(Urest, (leak)- Using these values of Iogn. we define a tuning curve that maps stimulus to synaptic conductance:
(s(s; Urest gleak) = (1 + @) (aa1nnyynaa (Urests gleak) e 3 (ITe)²
(72)
where Φ == 0.02 is used to compensate for the mean firing rate since Jeyn is determined according PF 0.1. The stimulus s is defined over the range from -90 to 90 to simulate orientation input. The corresponding tuning curve from stimulus to mean firing rate is obtained by applying TC through the empirically derived mapping from Jsym to μrr based on simulation data. In contrast to Padamsey et al.] (2022), we compute the full width at half maximum (FWHM) of the wrapped curve, as it may deviate from a Gaussian profile. To estimate the width along the optimal path, we fit the FWHM values using second-degree polynomial function in Urest and grekk and take the estimated values along the optimal states.
H.3 SPIKE TRAIN STATISTICS
In simulation, we input deterministic spike and probe the probability of responding spiking output under different configurations, including synaptic conductance, membrane leaky channel conduc- tance, and resting potential. This simulation characterizes the neuron excitability under different conditions. However, profiling this excitability curve does not directly provide the variance of out- put firing rate required in modeling Fisher information in our framework. To address this limitation, mathematically derived the variance of output firing rate based on the following assumptions:
The response output spikings are independent.
The input spikes follow Poisson distribution with a rate À E IR.
The variance of the spike count oZ be approximated by parabola of the mean par oz = MxdF (1 pr).
Based on these assumptions, we define the probability distribution of input spikes K as:
p(K- = 1:) (XT)*E-XC XI
(73)
k!
In other word, K Pois(XT), where T is given time window. Since our simulation input is single input, we can estimate the mean and variance of spike count F by the following empirical distribution:
p(F F = k; single spike input) = Pk. & [U)
(74)
It should be noted that the number of output spikes in the simulation is typically less than 3. Then the total output spike counts O is:
K F.
0 =
(by independence)
(75)
1=0
where F; is realization of F and F; corresponds to i-th spike in the input spiking train. Based on this model, we can calculate the expectation and variance of O by
[[O] [| = E[KE[F] = XT E[F] = XTμr,
(76)
Var[O] ([ + Var K]] = XTVar([F] XTE| XT + XT
(77)
where F[[S] and Var(S] be estimated from the simulation.
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Now, have approximate the relation between 76 and 77 formulate (77 in term of (76).
More specifically, want to
Var(O]
XT + 2T
E[O]
XT E[S]
XT (F) XT
(by of ~ n)p((1 - pr))
XT
(78)
1/ pr) + pe
Th |F (1xx 1)
Th (if uF small).
The last approximation holds when neurons operate the low spiking region, where make HF = 0.1 in our simulation. The eq. (78) also indicates a dispersed Poisson model. In eq. (78). we parabolic function to model the variance of spike count. As shown Fig. [ the fitted parabolas align closely with the simulation results, demonstrating the accuracy of the approximation.
simulation
o. 11pps(1 AP)
m V]
49
-73
0.2
0.1
0.0
<
0.2
80
0.2
Moat JaS. (cm"
0.0
0.2
100
0.1
0.0
+3
0.2
110
0.1
0.0
0.2
120
0.1
0.0
0.4
0.0 0.4060.8 PF
0.2. 0.000.40...0 0.00 pr
0.8
Figure 5: Fitted parabolas for the mean and variance of spike count. We fit a parabolic rela- tionship between the mean and variance of spike count across different cellular states (Wresty Steok). ) For visualization purposes, only every second value of Urest and Deak shown.
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H.4 FITTING OF WIDTH, ENERGY, AND NOISE
We fit the estimated width and energy in 2 and degree polynomial functions in Grest and Roaks respectively (see Fig. 6 Additionally, the estimated noise is fitted with
Pa(trests Pleak )e" "h Eveat + Wackkkk
(79)
where Pa denotes the degree-4 polynomial in Ureat and fleak and Wi and W2 are the learning weights. Note that we additionally impose non-negan non-negative constraint on the noise fitting parame- ters, further mitigating potential overfitting. The relative errors of these fittings are less than 2%. We select degree for energy fitting to enhance the stability of the numerical optimization eq. (1)) owing to the linearity of the constraint, which aligns with the straight energy contour. The degrees for the width and noise fitting are determined based on the maximum relative error.
I CHANGE OF ENERGY CONTOURS UNDER VARYING ACTIVITY LEVEL
The change of energy contours under varying k is shown in Fig. 7 Note that underlying noise is independent of but the angle of constant-energy lines changes, becoming steeper with higher cell activity. This results different curvatures of the optimal path shape.
J NOISE AND WIDTH CHANGE ALONG THE OPTIMAL PATH WITH VARYING ACTIVITY LEVELS
Here, we investigate how the activity level influences the density (defined as the inverse of tuning curve width) and the behavior of the noise dispersion curve (c(c)) along the optimal adaptations. Specifically, vary from 90 to 150 and fit the proposed models to the corresponding density and noise data as described in Sec.
The fitted parameters
shown in Fig- Bp-e. To characterize their dependence K, fit affine (see dotted lines Fig. 8). Due to the presence of outliers as
relation to (a1, 02) as a function 100, restrict the fitting data with x 100. The resulting fits (dotted lines in Fig, 8p, b) are given by:
(k) 6..93
10-11x K 1.462 10-8
(80)
(2(k) 1.626 10 4x 10- + 0.7452.
(81)
Additionally, we fit polynomial models of degree 2, 2, and to b, dg, and no. respectively (see dashed lines in Fig. (e-e). The fitted expressions are:
61 (x) 3.093 - 10 x 3.922 100x k + 2.013 10S,
(82)
by (xx) 5.999 102 + 1.888 10 k + 3.370 10
(83)
(K) -6.102 10-7 + 6.004 10- + 9.934 10
(84)
We further visualize the consequences of these deviations in the resulting density and noise disper- sion curves. As shown in Fig. [ g. the predicted density and noise dispersion values increasingly deviate from simulation results as & decreases. It should note that the values of ((K) can be easily derived from the values of ă and b.
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(A)
(b)
Width
Simulation
13
12
Relative [6]
108
033
40omk JuS [cme
104
100
-74
Preat [mn
70
65
Slook (cmm
120
-15
FWHM (dags]
-32
-78
106
Vreat
Total Energy étotal
(4)
120
15
Simulation
ATT//
4.0
130
3.5
65
3.0
laSton"
100
2.5
2.0
22
-14
-1
Voest Fm
Suak (u8) /m///mm
-10
120
Etotal
-34
2.0
[TTAATT/s]
Ureat [mV]
Noise 7}.
(
0.4
Simulation
LO5
0.2
Relative
1.03
Uhoa laS cmm?
0.1
1.01
774
0.0
Uved [mV]
[uS/an?]
120
1/4
1.01
1.02
1.03
1.04
1.05
-74
Ureat [mV]
Figure 6: Surface fitting to width, total energy, and noise. (a. b) We use 2 degree polynomial function in Grest and feak to fit the estimated FWHM, with the relative error provided. (c. d) We degree polynomial function in Grest and Gleak to fit the estimated total energy, with the relative error provided. (c. f) We use eq. (79) to fit the estimated total energy, with the relative error provided.
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1,056
110
1.045
juS e/m""
1.023
90
Slook
1,012
1.001
70
-66
-70
-74
-66
-70
-74
-66
-70
-74
[mV]
[mV]
[mV]
Figure 7: Impact cell activity level on constant-energy contours and optimal paths. We show varying angles of constant energy contours (white) resulting in different optimal path shapes (blue, orange, green) under x =90, 120, and 150, respectively.
1--9
0.270
0.765
www
0.760
90
100
110
120
130
140
150
90
130
110
120
130
140
150
R
(d)
300000
1.8
0.994
0.992
200000
1.7
0.990
1.6
770
90
100 110 120 130 140 150
90
100 110 120 130 140 150
90
100
110
120
130
140
150
K
K
(A)
(2)
(x))
d E
(0I(N) (1c) Capt + (x())))
Th
(x)) +
Eapt
(x)x)
0.98
[1 /wiidt
0.96
0.34
1.02
160
101
50
1.00
7hc
Capt [107 ATTP/4]
Capt
100 ATPP/4
Figure 8: Relationship between and the fitting parameters in the energy budget and the noise dispersion model as in eqs. (16) and (18). (a. b) To represent the energy budget in ATP/s units, we fit the parameters (01, (2) as an affine function of k. (c-e) To model the noise ATP/s units, we fit the parameters (bi, by, no) with polynomials of k. Dotted lines indicate polynomial fits of degree 2, 2, and 2 to b,, b2, and 7jo, respectively. (f, g) The resulting energy budget and noise dispersion curves (solid lines) generated using the fitted expressions are compared with the simulation data. For visualization purposes, we display every fourth value of x in the plotted results.
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K ANALYTICAL SOLUTION COMPARISON
We compare the our analytical solution to Ganguli Simoncelli (2010); Wang et (2016a) under the different objectives (see. Tabs. 5 to
Table 5: Comparison of infomax analytical solutions
Ganguli & Simoncelli (2010)
Ours
Wang et al. (2016a)
Density (tuning width)-
Ea R(s)-
Np(s)
Cp(s)
d(s)
Gain
Fa
R
g(s)
Fisher
Ea p(s)2
information
N²
C²p(s)²
FI(s)
The [)) R(+)²
Table 6: Comparison of discrimax analytical solutions
Ganguli & Simoncelli (2010)
Ours
Wang et al. (2016a)
Density
(tuning width)-1
Fa R(s) a+3 p(s) a+)
a+3
x Np(s)²
Cp(s)
d(s) Gain
α Ea R(s) a+3 p(s) a+3
Rp(s)
g(s)
Fisher
J Za
E (() a+3
information
x N2
CPp(s)
2a
FT(s)
Th (x) ((s)
a+3
Table 7: Comparison of general analytical solutions (Lp error, -29).
Ganguli & Simoncelli (2010)
Ours
Wang et al. (2016a)
Density
3-a
4-1 Np(s) 33-1
(tuning width)
Ea R(s) 33
p(s
3p
(p(s)T-21
d(s)
Gain
25
x E R(s)
33 38- p
x Rp(s) 1-33
g(s)
Fisher
Ea
Eap(8) 36-0
information
N2 Rp(s) 1-33
C2 1-23
F1(s)
7r (E)((s) 35
L DETAILS IN CONSISTENCY WITH DATA
L. DERIVATION OF THE PARAMETER SETTINGS
In Sec. 5 we aim to find the parameters for simultaneously accomplish 32% widening and 29% reduction in ATP consumption. Combining with the density analytical solution in model, the
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32% widening implies
Wms
detr
Ectr
Octrrr
+ 02
1.32
(85)
Wetr
dms
Ems
022
+
asterr
a2
0.19625,
(86)
O2
0.71
Ectr
Cetr
aitetr
where denotes metabolic stress.
Given a fixed tuning curve width derived from the fixed population size, we reduce the firing rate according (R) in Ganguli & Simoncelli (2010) according to the decrease in the maximum mean firing rate in the mouse neuron. Due to the gaussian assumption, 32% widening, and a uniform prior, we caleulate the decrease in the maximum mean firing rate
max 1.32g 2
320)
76% (1
24%
(87)
1.32
max, v/
C
decreuse
Similarly, due to 32% widening and uniform prior, can calculate the decrease in the coding capacity in Wang et al. (2016a) by
1
Wetr
dmss
Cms
Cms = (1- 24%
Cetr-
(88)
1.32
Wms detr
Cetr
decreeue
L.2 MEAN FIRING RATE DEVIATION
Here, we analyze changes in firing rate under different coding objectives and prior distributions (see Fig. 9b. Across all coding objectives, we adopt a consistent configuration: b/ b/(ueatr) = 0.19625,
29% reduction in energy expenditure, and 1, consistent with the settings used in Sec. [ Addi- tionally, we fix E and R(s) for all cases, as these parameters yield a control tuning width of 35 degrees under uniform prior following the simulation conditions reported in Paadassy ett al. (2022). After generating the population codes according to Tab. [ we extract the tuning curve corresponding to the neuron with preferred stimulus s, 90% Each extracted tuning curve is nor- malized by the peak value of its respective control condition, a transformation that does not affect the analysis of firing rate deviation.
Fig. 9h shows the uniform prior assumed in Sec. 5 along with the infomax case (Fig. Pp reiterates objective functions lead the same original tuning curves and adaptations, all with perfect homeostasis.
Fig. 4p). Here we illustrate that, under constant R(s)
In addition to the uniform prior, we also consider as non-uniform prior the distribution of edge orientations in natural scenes, as reported in Girshick et al. (201T). When replacing the uniform prior with the non-un non-uniform one, the deviation in mean firing rate (compared to each control condition, shown in dotted purple) remains within .2% across infomax, discrimax, and Lı error objectives. As shown in Fig. D3 and h, the tuning curves deviate from Gaussian profile due to the dependence of the gain function (s)) on the prior distribution for 4p norm losses (sce Tabs, 6 and
L.3 MEAN FIRING RATE DEVIATION WITH NON- NON-GAUSSIAN TUNING CURVE
Although we adopt the Gaussian function as our base shape i, our model is not restricted to it. To further generality, evaluate the homeostasis approximation error using one-dimensional Gabor function, a commonly adopted tuning curve (Manning et al., 2024). As shown in Fig. I0] the mean FR deviation remains within 5%, demonstrating that our homeostasis approximation also holds for non-Gaussian tuning base shapes. Note that the optimality of the derived Gabor tuning curve still requires evaluation of the tiling property, as discussed in Sec.
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Food-restrieted
Infomax
Discrimax
Normalized Mean FR
1.4
1.0
1.0
0.8
0-8
0.8
Probabiity
0.0057
248
0.5
0.8
0.008
0.0035
24
0.4
0.4
23
0.2
0.2
0.0053
30
0.0
0.0
120
1.00
50
100
15c
(4)
(P
(h)
Normalized Mean FR
0.0075
Profablity
0.0070
0 0.085
0.0050
0.0045
100
150
30
100
150
30
150
100
150
(deg)
(deg)
Stimulus (deg]
[deg)
Subplot Label
(b)
(c)
(d)
(f)
(g)
(h)
Mean FR deviation
-0.14%
-0.14%
-0.14%
-0.20%
-0.16%
-0.17%
Gaussian base shape adaptation under varying objectives and non-uniform prior results in slight violations of homeostasis. (a) The uniform prior used in Fig. 4 (b) Under uniform prior and the infomax objective, reiterates Fig. 4b. (c) Under the discrimax objective, optimal gain varies function of p(s) and R((s). However, by assuming uniform p(s) and R((x), the resulting tuning curve remains unchanged from the infomax case. (d) The I, error minimization objective yields the same results to (c). (c) A non-uniform prior derived from the distribution of edge orientations in natural scenes, as reported in Girshick et al. (201T). (f) The resulting tuning curve under the infomax objective with the non-uniform prior. (g. h) Under the discrimax and general objectives, the optimal gain varies with p(s) (and R(s), though we assume a constant R in this simulation). Despite these variations, the mean firing rate deviation remains within 2%. (i) The mean FR deviation in the above conditions.
Figure 9:
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Control
Food-restrieted
Infomax
Discrimax
Normalized Mean FR
1.0
1.2
0.8
0.5
Probabiity
0.0057
0.8
0.8
0.008
0.0035
0.4
0.4
0.4-
0.2-
0.2
0.2-
0.0053
0.0
0.0
0.0
(4)
(h)
Normalized Mean FR
0.0075
Profubiiity
0.0070
0 0.085
0.0050
0.0045
100
150
100
150
100
150
30
300
Stimulus (deg)
Stimulus (deg)
Stimulus [deg]
Stimulus (deg)
Subplot Label
(b)
(c)
(d)
(f)
(g)
(h)
Mean FR deviation
-0.3 1%
-0.31%
-0.3
-0.45%
-0.37%
-0.39%
Gabor base shape adaptation under varying objectives and a non-uniform prior results in slight violations of homeostasis. (a) The uniform prior used in Fig. 4 (b) Under uniform prior and the infomax objective. (c) Under the discrimax objective, the optimal gain varies as function of p(s) and R(s). However, by assuming uniform p(s) and R(s),, the resulting tuning curve remains unchanged from the infomax case. (d) The L error minimization objective yields the same results to (c). (c) A non-uniform prior derived from the distribution of edge orientations in natural scenes, as reported in Girshick et al. (20IT). (f) The resulting tuming curve under the infomax objective with the non-uniform prior. (g. h) Under the discrimax and general objectives, the optimal gain varies with p(s) (and R(s), though we assume a constant
Figure 10:
this simulation). Despite these
variations, the mean firing rate deviation remains within .%. (i) The mean FR deviation in the above conditions.
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M FITNESS OF THE DERIVED TUNING CURVES
Here, we adopt the likelihood method to evaluate the fitness of our model on the data presented in Padamsey et al. (2022) and compared to Ganguli & Simoncelli (2010); Wang et al. (2016a).
M.1 LIKELIHOOD
Following the dispersed Poisson distribution, we can derive the log likelihood function of a single neuron
log L (s)(r) (s), r (s),
(s))
(89)
k
(r1) - (s))²
= log
exp
(90)
2 (E) (n)))))
hen
3=1
(r) - An (s))
log
exp
(91)
2 (E)bn(s)
2n (1) (9)
5=1
(T)) to (s)
27
k
1
log (2TTe(E))
(92)
2
2/ (E)h
j=1
log (2TT (F)))n)))) (8))
(0) (2) ((s))-h,(0 (s) An((s)
(93)
2
2m E h
f=1
where & is the total number of responses (trials). For optimization, we can drop the constant terms, giving:
( (8) - 1s) 2
log Ly
log (+))
(94)
2
(F)())) ())))
To extend the model to the entire population of N neurons, follow the standard assumption that the noise in each neuron's 's response is independent. This assumption allows us to formulate the total log-likelihood for the population the sum of the log-likelihoods for each individual neuron.
N
log Crotal
log Lu
(95)
n=1
Note that future work could explore the impact of information -immininnimiiiiggggg noise correlations (Moreno- Bote et al. 2014), but this would require grounding in data that has never been collected to model how these correlations might be impacted by metabolism.
M.2 CALIBRATION
Estimating the gain function y and density function d by maximizing the log-likelihood in eq. 995) challenging. The difficulty arises because the second term in the RHS of eq. (94). has a complex, non-linear dependence on y and d. Furthermore, the optimization is constrained by the requirements of approximate homeostasis and energy constraint. However, we believe that this method can be used to calibrate the parameters E, and R(s).
ΣØ= (s)
log Ln x - log (s()))(((>>- - D(sw))) -((s))))))
(96)
((D)))))(())) -D(s.)
Based on the analytical solution (assuming an infomax objective) presented in the manuscript, can substitute g(s) and d(s) with:
g(s) = E//a
(97)
d(s) E!/a R(s) R1ss))((s)).
(98)
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Although eq. (96) is defined in terms of D rather than d, can approximate D numerically. We then can exploit the differentiability of the involved functions to iteratively optimize the parameters using gradient-bart basedd approaches. Nevertheless, due to the nonlinear nature of eq. (96), the optimization process may converge to suboptimal local minima.
By the approach above, can integrate analytical solution with data-driven calibration of the parameters E, at and R(s). However, the stability and efficacy this method have not yet been rigorously evaluated. We believe this presents an interesting direction for future work, especially as larger datasets become available.
M.3 LIKELIHOOD ON BIOLOGICAL DATA
We can apply the method in eq. (96) to evaluate the fit of the tuning curves shown Fig. 4 to the data in PPadamsey al. (2022) Note that this data has significant drawbacks: rather than tak- ing repeated measurements from the same cell, then repeating the full experiment under a tighter metabolic constraint, Padamsey et al. (2022) instead aligns and averages responses from 29 cells in the control condition and 32 different cells the food restricted condition, which are taken as representative of variations over repeated trials of a single underlying tuning curve. Because this is the only data available, we will use as an illustrative example here, but note that future spike train data is needed refine these estimates. Specifically, find log likelihoods as -155 (Ganguli & Simoncelli 2010). -107 (Wang et al. 2016a). -85 (ours), -47 (ours with optimized 1). This shows the expected ordering of fit quality.
N EVALUATION OF FLATTENING EFFECT ON NON-INDEPENDENT NOISE
Our proposed framework assumes conditionally independent noise (see Sec. A for analytical tractability. However, this assumption may be unrealistic due to the presence of noise correlations across neurons. To assess the extent to which such noise model mismatch induces a flattening effect, we incorporate differential correlations (Moreno-Bow et al. 2014), which have been identified as dominant factor limiting information transmission.
Specifically, we aim to optimize the following constrained optimization problem corresponding to the infomax case:
argmax / p(s) log s()),(())
(B T/ ())))))(())((())
ds,
(99)
s.t. p(s)g(s) R(s)d(s),
(100)
J pss)
(101)
where e 2 0 denotes the strength of differential correlations. the simulation, we set 17 (E) 1, as its value varies only slightly (from to 1.06; see Fig. 33. For simplicity, we further specify R(s) = 1. The stimulus distribution p(s) is chosen to match the empirical distribution of edge orientations in natural scenes reported by Girshick et al. (2011). However, numerically solving this constrained optimization problem challenging. To mitigate this difficulty, we use the approximate homeostasis constraint to express d(s) in terms of g(s) and relax the energy constraint using an form regularization term with penalty weight Apenalty = 100. The relaxed optimization problem is as follows:
p(s) / p(s) log [+cp(s)'''s))
ds Apenatry| || / p(s)g(s)) E1I
argmax 9(-)
(102)
We then perform gradient descent with step size of 10-3 for 3000 iterations. After obtaining the optimal g(s), we recover the optimal d(s) by the approximate homeostasis constraint. Note that vary the correlation strength up to 1000, which is substantially larger than the noise variance of individual neurons (i.e., 77x 1).
The optimal gain and density functions exhibit only modest changes across different levels of differential correlations; in contrast, metabolic stress produces a substantially larger effect (see Fig. [Tp-b). We then derive the optimal tuning curves associated with the gain and density functions
33

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Published as a conference paper at ICLR 2026
in Fig. [ b, which display the same flattening pattern predicted by analytical solution in the infomax setting. Fig. [Td, we quantitatively evaluate the relative broadening for each matched pair of correlation strengths in the control and food-restricted conditions. Although increasing the differential correlation strength from O to 1000 results in an additional 4% broadening, this effect is minor compared to the impact of metabolic stress, which produces approximately 38% broadening.
This numerical experiment demonstrates that the flattening effect predicted by our proposed model remains robust even when differential correlations are taken into account.
We conducted similar experiments with Apemalty = 10 and Apenalty 1000 (see Figs. 12 and 13) to evaluate whether the broadening effect arises from any numerical artifact introduced by the choice of the penalty weight Apenalty. As shown in Figs. to I3 the trend of the broadening effect remains consistent increasing the differential correlation strength results in additional broadening. We therefore conclude that the broadening effect is not the numerical artifact.
Additionally, we computed the regularizer error,
// p(ss)
under Apenalty 10, 100, 1000, as shown in Fig. 14
The results show that the default setting 100 is reasonable choice for solving the relaxed problem.
Apenalty
(4)
1.00
0.008
Control cases (E-1)
0.95
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in food-restricted case
Figure 11: Comparison of gain, density, tuning curves, and broadening effects under vary- ing levels of differential correlations. (a-b) Optimal gain and density functions obtained across different strengths of differential correlations. Blue and red curves represent the control and food- restricted conditions, respectively, with color gradients indicating increasing correlation strength. (c) Optimal tuning curves corresponding the gain and density functions shown in panels (a) and (b). (d) Relative broadening effect for each matched pair control and food-restricted conditions. The top-left pair corresponds the case assuming independent noise. Note that this experiment is under Apemalty 100.
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Published as a conference paper at ICLR 2026
Apenalty = 10
(9)
(b)
1.1
0.008
1.0
Gain
Control cases (E-1)
Density
0.9
0.006
0.8
0.004
Food-restricted cases (E-0.76)
45
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90
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180
Stimulus [deg.]
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(6)
(4)
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Normalized HR
in control case
0.77
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.99
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359.7 38
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in food-restricted case
Figure 12: Comparison gain, density, tuning curves, and broadening effects under vary- ing levels of differential correlations. (a-b) Optimal gain and density functions obtained across different strengths of differential correlations. Blue and red curves represent the control and food- restricted conditions, respectively, with color gradients indicating increasing correlation strength. (c) Optimal tuning curves corresponding to the gain and density functions shown in panels (a) and (b). (d) Relative broadening effect for each matched pair of control and food-restricted conditions. The top-left pair corresponds to the case assuming independent noise. Note that this experiment is under Apenally 10.
35

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Published as a conference paper at ICLR 2026
1000
penalty
(4)
(b)
1.00
0.008
Control cases (E=1)
0.95
0.007
Gain
0.90
Density
0.006
0.85
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0.80
Food-restricted cases (E=0.76)
0.004
0.75
45
90
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45
180
Stimulus [deg.]
Stimulus [deg.]
(6)
1.0
0.00
1.396
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control case
0.28
Relative broadcning
0.77
1.394
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1.392
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0129.15
1.390
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16
68
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42
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in food-restricted case
Figure 13:
Comparison of gain, density, tuning curves, and broadening effects under vary- ing levels of differential correlations. (a-b) Optimal gain and density functions obtained across different strengths of differential correlations. Blue and red curves represent the control and food- restricted conditions. respectively, with color gradients indicating increasing correlation strength. (c) Optimal tuning curves corresponding to gain and density functions shown in panels (a) and (b). (d) Relative broadening effect for each matched pair of control and food-restricted conditions. The tof-left pair corresponds to the case assuming independent noise. Note that this experiment is under Apenalty 1000.
Control Case (E=1) 1)
Food-restricted Case (E=0.76)
3.00%
3.00%
Aponatt: 10
2.50%
2.50%
Apenaty: 100
plejats) EI
2.00%
Aponany: 1000
d( maa
2.00%
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C
Figure 14: Evaluation of the effect of penalty weight Apenally.
36


Reviews:
[{"review_id": "ApDcFp0z0f", "reviewer": "Reviewer_iz34", "summary": "The idea that brain representations are fundamentally shaped not only by input statistics but also on metabolic constraints has proven very important for our understanding of sensory coding. Nonetheless traditional efficient coding models don;t capture the exact nature of the constraints which leads to incorrect predictions in the low energy regime compared to recent experimental measurements (Padamsey et al., 2022). The paper introduces a new way of expressing these energetic constraints that is mathematically tractable and biophysically well justified, part of a general framework that has past model as special cases, and accounts for the experimental observations that demonstrate increased noisiness as the price of more stringent energy constraints.", "questions": "Neural results and discussion: i would have liked to see some concrete predictions that the model makes and perhaps some comments of how the energy-dependent encoding process affects processing downstream\n\nsome additional validation of results in simulations, e.g. documenting how other coding considerations affect the degree of flattening would make the coding numerics more substantial.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Clean mathematical formalism for coding efficiency with new constraints.\n\nBiophysical simulations that link mechanistic considerations with coding level abstraction. \n\nExplains for the first time recent experimental observations on the effect of limited energy availability on neural coding.", "weaknesses": "While i do enjoy the mathematically clean formulation, the change in the constraint is in and of itself an incremental contribution at the technical level. \n\nThe link to data and discussion sections are very brief and need expansion.\n\nNumerical results are very minimal.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "OCKOeKYHti", "reviewer": "Reviewer_LgZi", "summary": "The paper tackles the adaptation of tuning curves in optimal networks in situations of metabolic stress. Authors model the observation that under metabolic stress, tuning curves of neurons in the mouse visual cortex show flattening while the average firing rates are maintained. A theoretical framework describing tuning curves is developed that captures such flattening of tuning curves in situation of energy limit that affects noise levels in single neurons. Simulation of biophysical neurons are carried out to show that the mechanism suggested by the theory can be also seen in the biophysical model of single neurons.", "questions": "1) What kind of neuron model is used in simulations? I believe it is a Hodgkin-Huxley model, but this is not specified.\n2) In line [300], authors mention mean spike count of 0.8 and 0.2. It is unclear to me what are the units used here?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The theoretical framework as well as numerical simulations seem technically sound and carefully conducted (even though I had not checked all the details). The question attacked by the paper is  seldom explored and is of biological relevance. Some of the relevant weaknesses are appropriately discussed.", "weaknesses": "1) Main weakness is the assumption of conditionally independent firing rates. Cortex, including the visual cortex of the mouse, consists of highly recurrently connected networks of Excitatory and inhibitory neurons that have strong influence on each-other's activity (see for example Chettih and Harvey, Nature 2019). I am not convinced that the assumption of independent neurons can bring crucial insights about the neural code in the cortex, including the study of tuning curves. Can authors comment why do they think that a model with independent neurons is relevant?\n\n2) There is a lack of citations on several occasions. \nExamples:\n[line 50 ] Authors mention \"Simple models of energy constrains\" but not cite any so it remains unclear which models they are referring to. \n[line 54] ...\"previous population coding models capture either the shortening or the widening of energy-limited tuning curves, but not both\". Which previous models show this and why are they not cited at this point? \n\n3) Authors situate the current work among the existing literature in a way that seems rather problematic. In line [122] authors comment on the importance of the activity of the sodium/potassium pump for the metabolic expenditure of neurons. The activity of the sodium/potassium pump seems to be a direct consequences of spiking activity, and the maintenance of the reversal potential also depends on neuron's firing rate. It can be argued that models that formulate the metabolic cost that is dependent on the firing rate or the number of fired spikes nevertheless capture, even though indirectly, sources of metabolic expenditure that are biologically highly relevant. Such efficient models simply do not model the situation of important metabolic stress, but instead use the metabolic cost as a means to regulate the activity levels in the network and constrain the solution appropriately (in a general situation without energy deprivation). While a state of energy deprivation seems an important constraint from the  evolutionary perspective, I am not convinced that neural coding would only adapt to this particular constraint.\n\nRecent research (Gutierrez and Deneve, eLife 2019, Koren et al., eLife 2025) has discussed the role of metabolic efficiency in optimal spiking networks. Both studies use a metabolic constraint that is formulated as a cost on the number of spikes fired. The first study (Gutierrez et al.) found a formulation of efficiency that gives rise to a transient adaptation in single neurons on a time scale of seconds. The second study (Koren et al.) found that a number of biophysical parameters in primary cortex can be captured when the metabolic efficiency is taken into account, and that metabolic efficiency was shown to have an important effect on optimal coding solutions. Can authors comment on their results in light of these two studies and also touch on the difference in the use of the metabolic efficiency in their model compared to the one in papers mentioned above? \n\n4) The paper is difficult to read and could use a rewrite with focus on clarity. I believe bringing in some more simplicity when possible would make the paper much more appealing. In particular, the first paragraph of discussion gives a good intuitive motivation for the model and I suggest to move this part earlier in the paper, possibly to the introduction.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "WDvx3Q3dVt", "reviewer": "Reviewer_rMx9", "summary": "The authors introduce a new method for modelling the interaction of metabolic stress and tuning curves. They show that different from previous models, their model is capable of reproducing the experimental findings reported in (Padamsey et al., 2022) - mainly that under metabolic stress, tuning curves widen and reduce in height. To derive model parameters, they hypothesise that cells adjust neuron parameters (rest potential, leak conductance) to minimise noise levels under varying energy levels.", "questions": "1. As far as I can see, the main novelties are: 1) The activity and energy-dependent dispersion, 2) The generalised energy constraint, although in the derived model, alpha = 1 is used which aligns with previous models. 3) The fit to simulated results (with energy estimates tying neural activity to ATP needs) to  obtain alpha and eta. Is that correct?\n2. Eq. (1) is for Poisson neurons? A citation would be helpful.\n3. What exactly is kappa? How is it related to R(s), R_n, R, etc.? Also, how is R(s) determined?\n4. Eq. (7): E is stated to be energy. However, since this expression is valid for any alpha >= 1, this is weird unit-wise (and in fact, for alpha = 1, the authors identify E as the mean rate R). \n5. The final model is more complex than the two comparison models (Ganguli & Simoncelli; Wang et al.), and parameters are actually fit - not on experimental data, but using simulations. So it seems like increased expressivity is paid for by an increase in complexity. Could you comment on how well you expect this to generalise to other experimental setups? In particular, does your model predict any other effects?\n6. If I understand correctly, eta_k is obtained for different combinations of v_rest and g_leak. For activity kappa and a given energy level, you then determine the values of (v_rest, g_leak) that minimise eta_k (which is later used to fit a function for eta_k)? This part was quite dense to read, so a few more explanations might be helpful here.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The presented model is a simple extension of previous ones, which can be recovered by setting the function eta and parameter alpha accordingly.\n2. Analytical solutions are provided for their model for different optimisation objectives.\n3. They use simulations inspired by the experimental protocol used in (Padamsey et al., 2022) to fit alpha and eta in a biologically grounded way. This is the main difference between previous models and the proposed one, leading to alpha = 1 (which has also been used in Ganguli & Simoncelli, 2010), and a functional relationship for eta.\n4. Their model faithfully reproduces experimental findings. They compare this with two alternative models, which fail to fully capture experimental observations. In particular, only their model shows firing rate homeostasis.", "weaknesses": "1. Since the introduced model is calibrated using simulations, it is not clear how well it will generalise to other experimental setups.\n2. In general, the part about extracting eta_k and alpha from simulations is quite dense and hard to follow, which could be improved.\n3. Some of the used variables have to be explained better, as it is not evident what they are or how they are determined (see questions).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)