---
title: "Statistical physics of neurons: from ion channels to avalanches"
description: "From the membrane capacitor and the Hodgkin–Huxley spike to single-channel noise and neuronal avalanches, and what avalanche statistics can and cannot show."
canonical_url: "https://sandboxphysics.com/stories/neural-statistical-physics/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/neural-statistical-physics/"
last_modified: "2026-10-31"
---

[Statistical physics of neurons](https://sandboxphysics.com/stories/neural-statistical-physics/)

FR42 · An interactive course

# Statistical physics of neurons: from ion channels to avalanches

Sandbox Physics · 2026-10-31

A nerve impulse is a membrane capacitor whose voltage sodium ions push up and potassium ions bring back down. This course follows that picture from one patch of squid axon to a sheet of a thousand model neurons. Recomputing the 1952 equations of Hodgkin and Huxley reproduces almost every entry of their spike tables. Counting the channels one by one turns the threshold into a probability. Connecting neurons into cascades raises a harder question: can avalanche statistics show that a network is critical? In simulations where the answer is known, a power law alone cannot.

[Start with the membrane](https://sandboxphysics.com/stories/neural-statistical-physics/the-membrane-as-a-capacitor/) Open an experiment

The membrane as a circuit. The lipid layer stores charge like a capacitor $C_M$. Each kind of channel is a conductance $g_x$ in series with a battery at its reversal potential $E_x$; the arrows mark the sodium and potassium conductances that change with voltage. An applied current $I_{\mathrm{ext}}$ either charges the capacitor or leaves through the channels. Original schematic.

## From one patch of membrane to a recording of many neurons

The first chapter treats the membrane as a capacitor with leaky, battery-backed channels and works out why the currents cancel at rest. The second holds the voltage fixed to separate the sodium and potassium currents and introduces the gating variables. The third puts them back into the membrane equation and recomputes the threshold and the spike tables of Hodgkin and Huxley. The fourth turns each gating variable into the probability that one channel is open, and the fifth follows the resulting noise into blurred thresholds, scattered spike times and spikes that fire on their own. The sixth connects many neurons into a branching network and asks what an electrode array that sees only part of it records. The seventh tests whether a power law of avalanche sizes can show that a network is critical. The last chapter is a guided reading of the 1952 paper.

You need the capacitor and Ohm’s law from introductory electricity, first-order differential equations, the binomial distribution and reading a log–log plot. The Boltzmann factor, Markov chains, maximum likelihood and the regression estimators are introduced when they are first needed. The course stays with model membranes and model networks: it does not reconstruct a brain, diagnose anything or say what consciousness is. [FR42](https://github.com/yangzichao/sandboxphysics/issues/72)

1. 01 · L1

   ## [The membrane stores charge like a capacitor](https://sandboxphysics.com/stories/neural-statistical-physics/the-membrane-as-a-capacitor/)

   A cell membrane is a capacitor with leaky, battery-backed channels. Find how much charge moves the voltage and why the currents cancel at rest.

2. 02 · L2

   ## [Voltage clamp separates the sodium and potassium currents](https://sandboxphysics.com/stories/neural-statistical-physics/voltage-clamp-and-gates/)

   Fix the membrane voltage and the capacitive current vanishes, leaving the ionic current. Three gating variables then give the sodium peak after a step to 60 mV.

3. 03 · L2

   ## [The action potential from one equation, all or none](https://sandboxphysics.com/stories/neural-statistical-physics/the-action-potential/)

   Put the gates back into the membrane equation and sodium feedback makes a spike. Find the threshold between 6 and 7 nC/cm² and recompute Tables 4 and 5.

4. 04 · L3

   ## [Splitting the gating variables into single channels](https://sandboxphysics.com/stories/neural-statistical-physics/channels-one-at-a-time/)

   A gating variable is a probability and a channel is a small Markov chain. The open count is binomial, and one opening lifts a small patch by 0.2 mV.

5. 05 · L3

   ## [Channel noise blurs the threshold and the spike time](https://sandboxphysics.com/stories/neural-statistical-physics/channel-noise-and-spike-timing/)

   Repeat one shock on a small patch: the threshold becomes a probability, spike times scatter and noise fires spikes. When does the square-root law return?

6. 06 · L3

   ## [Neural avalanches from branching spikes, seen through a partial recording](https://sandboxphysics.com/stories/neural-statistical-physics/from-neurons-to-a-recording/)

   How many spikes does one spike trigger? At the critical point cascade sizes fall as s to the minus 3/2. Then see what recording only part of the network does.

7. 07 · L4

   ## [Why a power law cannot prove the brain is critical](https://sandboxphysics.com/stories/neural-statistical-physics/when-a-power-law-misleads/)

   Fit a power law properly, then watch uncoupled models pass the test. With the truth known, only perturbing the network separates critical cascades from mimics.

8. 08 · L4

   ## [Reading Hodgkin and Huxley 1952 with questions in hand](https://sandboxphysics.com/stories/neural-statistical-physics/reading-hodgkin-and-huxley-1952/)

   A reading guide to the 1952 paper: translate its signs, find the equations, constants and method by page, check Tables 4 and 5, then read the authors’ caveats.

## Experiments along the way

[01 **Fire a squid axon** Shock a patch of squid axon membrane, watch sodium rush in and potassium follow, and find the charge that separates a spike from a ripple.](https://sandboxphysics.com/labs/hodgkin-huxley-clamp/) [02 **Count the channels** Shrink the patch until a few thousand channels decide each spike, and watch the threshold turn into a coin flip and spike times scatter.](https://sandboxphysics.com/labs/stochastic-ion-channels/) [03 **Record an avalanche** Tune how many neighbours each spike excites, record part of the sheet with an electrode array, and test whether the cascades are really critical.](https://sandboxphysics.com/labs/neural-avalanche-criticality/)

The squid-axon experiment integrates the 1952 equations with the constants of the paper; its recording trace adds simulated amplifier noise, drawn separately from the model trace. The channel experiment treats the sodium and potassium channels of a patch as independent Markov chains, counting how many sit in each state, and repeats each shock many times. The avalanche experiment runs a simulated sheet of 1000 units whose coupling you set, so the truth behind every analysis is known. None of the three analyses a real neural recording.

For thermal noise on a single bead and Markov chains driven out of equilibrium, see the companion course [Stochastic thermodynamics: work, heat and information for one small system](https://sandboxphysics.com/stories/stochastic-thermodynamics/).

## Sources and what we recomputed

1. [A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, Journal of Physiology 117, 500–544 (1952), PMC1392413](https://pmc.ncbi.nlm.nih.gov/articles/PMC1392413/)

   We read every page, pp. 500 to 544, from the scanned pages in PubMed Central. Sign convention on pp. 505 and 506: V is the displacement from rest with depolarisation negative, and inward current is positive. Rate constants: Eqs. (12), (13), (20), (21), (23) and (24). Eq. (26) and Table 3 on pp. 518 to 520: the full membrane equation and its constants at 6.3 °C, with the leak potential chosen so that the current is zero at rest. Numerical method on p. 523. Figs. 12, 13 and 21: threshold between 6 and 7 mV. Table 4 on p. 527: spike shape. Table 5 on pp. 532 and 533: sodium and potassium movements. Fig. 20: refractory period. Fig. 22: anode break. P. 541: their warning that the equations are an empirical description. We recomputed Tables 4 and 5 for the membrane action potentials, not the propagated one.

2. [W. Gerstner, W. M. Kistler, R. Naud and L. Paninski, Neuronal Dynamics, Cambridge University Press (2014), online edition, Sections 2.1 and 2.2](https://neuronaldynamics.epfl.ch/online/Ch2.S1.html)

   Section 2.1: the Boltzmann factor, Eq. (2.1), and the Nernst potential, Eq. (2.2), with the sodium example near +67 mV and the potassium example near −83 mV. Section 2.2 (Ch2.S2.html): charge conservation on the membrane, Eqs. (2.3) and (2.4), the three ionic currents, Eq. (2.5), first-order gating, Eq. (2.6), and the response to a voltage step, Eq. (2.7). This book writes u as the membrane potential with depolarisation positive, the convention our course uses.

3. [J. H. Goldwyn and E. Shea-Brown, The what and where of adding channel noise to the Hodgkin–Huxley equations, arXiv:1104.4823v1; PLoS Computational Biology 7, e1002247 (2011)](https://arxiv.org/abs/1104.4823v1)

   Version 1, 25 April 2011, read in full. Eq. (1): the Hodgkin–Huxley equation with depolarisation positive. The Markov-chain description of independent channels and its deterministic limit. Eqs. (10) to (13): for a fixed voltage history, the number of open channels is binomial with mean and variance set by the gating variables. Figs. 1 to 3: 100 µm² holds 6000 sodium and 1800 potassium channels, 10 µm² holds 600 and 180, with a spontaneous spike from channel noise alone. Their comparison of current, subunit and conductance noise. The label of Eq. (13) in v1 says sodium where the formula is the potassium variance.

4. [S. Zapperi, K. B. Lauritsen and H. E. Stanley, Self-organized branching processes: mean-field theory for avalanches, arXiv:cond-mat/9509161v1; Physical Review Letters 75, 4071 (1995)](https://arxiv.org/abs/cond-mat/9509161v1)

   Version 1, read in full. The binary branching process with critical probability one half. Eqs. (6a) and (7): the generating function of avalanche sizes. Eq. (8): the size distribution with exponent 3/2 and cutoff size −2/ln[4p(1−p)]. Eq. (11): the critical distribution. Lifetimes with exponent 2. We recomputed the exact small sizes from the Catalan numbers and checked the asymptotic prefactor.

5. [J. Wilting and V. Priesemann, Inferring collective dynamical states from widely unobserved systems, arXiv:1608.07035v3; Nature Communications 9, 2325 (2018)](https://arxiv.org/abs/1608.07035v3)

   Version 3, main text pp. 1 to 3; the supplementary methods were not read. The branching process with drive, the conventional estimator as the slope of the next count against the current count, the result that subsampling multiplies every lagged slope by the same factor, and the multistep regression estimator that fits that exponential across lags. Their examples: a process with m = 0.9 sampled at 10 % gives a conventional estimate of 0.312.

6. [V. Priesemann and O. Shriki, Can a time varying external drive give rise to apparent criticality in neural systems?, PLoS Computational Biology 14, e1006081 (2018)](https://doi.org/10.1371/journal.pcbi.1006081)

   Introduction, Results and Discussion read; Methods not read. How avalanches are defined from binned events. A Poisson process with a constant rate gives no power law; one whose rate switches among a few levels gives sizes close to a power law with exponent near 3/2 over a limited range. The measures that tell the two apart, and their conclusion that passive analysis cannot rule out a drive that mimics critical activity, so a small controlled perturbation is needed.

7. [A. Clauset, C. R. Shalizi and M. E. J. Newman, Power-law distributions in empirical data, arXiv:0706.1062v2; SIAM Review 51, 661–703 (2009)](https://arxiv.org/abs/0706.1062v2)

   Version 2, read in part: the abstract, Box 1, Section 3.1 on discrete maximum likelihood, Section 3.3 on choosing the lower cutoff by the Kolmogorov–Smirnov distance and the p-value parts of Section 4. Discrete approximation for the exponent with the half-integer shift; p above 0.1 means plausible, not proven; a large p does not show that the data follow a power law.

8. [J. Touboul and A. Destexhe, Can power-law scaling and neuronal avalanches arise from stochastic dynamics?, arXiv:0910.0805v3; PLoS ONE 5, e8982 (2010)](https://arxiv.org/abs/0910.0805v3)

   Version 3, read in part: the abstract, introduction and part of the methods. A straight line on a log–log plot is not a robust test; thresholded stochastic signals and shuffled surrogates can show similar scaling.

9. [M. C. Morrell, A. J. Sederberg and I. Nemenman, Latent dynamical variables produce signatures of spatiotemporal criticality in large biological systems, arXiv:2008.04435v1; Physical Review Letters 126, 118302 (2021)](https://arxiv.org/abs/2008.04435v1)

   Version 1, read the abstract and the model section. Neurons that fire independently given a few slowly varying shared inputs reproduce the scaling under coarse-graining that had been read as a sign of criticality.

10. [D. P. Carcamo and C. W. Lynn, Emergence of criticality in models of real neurons, arXiv:2609.09438v1 (8 September 2026), preprint, not peer reviewed](https://arxiv.org/abs/2609.09438v1)

    Version 1, read in full. Models fitted to recordings of many neurons tend to sit near critical points because a small region of parameters maps onto a large region of possible statistics; the effect grows with the number of neurons. Their mean-activity model, Eq. (3), gives a double-well approximation for the fitted field and coupling. They report that it converges to the exact parameters for about 100 neurons or more. We solved the same inverse problem exactly and compared. On p. 11 the text says the inferred couplings exceed the critical strength and writes this as a coupling of at least 0; the critical strength of this model is 1, so we read it as at least 1.