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Spikes can transmit neurons' subthreshold membrane potentials

Valentin Schmutz

TL;DR

The study addresses whether spikes inherently prevent transmission of subthreshold membrane potential fluctuations and shows that, in a high-dimensional setting, such information can be perfectly transmitted. Presynaptic neurons are modeled as N linear-nonlinear-Poisson units with a monotone transfer function; their subthreshold potentials are assumed to be a stationary Gaussian vector with a low-rank, approximately weakly correlated covariance generated from random Gaussian features; a postsynaptic population reads out membrane potentials by a dense linear decoder whose weights are proportional to the presynaptic covariance, enabling exact reconstruction in the large-N, large-P limit with P much smaller than N. The result holds for a broad class of transfer functions, including discontinuous ones, and relies on concentration of measure in high dimensions, implying that the nonlinear spiking nonlinearity becomes immaterial in this limit. Biologically, this predicts that subthreshold information can be recovered in desynchronized, high-dimensional cortical states, and suggests experimental tests using voltage imaging to compare reconstruction quality across brain states.

Abstract

Neurons primarily communicate through the emission of action potentials, or spikes. To generate a spike, a neuron's membrane potential must cross a defined threshold. Does this spiking mechanism inherently prevent neurons from transmitting their subthreshold membrane potential fluctuations to other neurons? We prove that, in theory, it does not. The subthreshold membrane potential fluctuations of a presynaptic population of spiking neurons can be perfectly transmitted to a downstream population of neurons. Mathematically, this surprising result is an example of concentration phenomenon in high dimensions.

Spikes can transmit neurons' subthreshold membrane potentials

TL;DR

The study addresses whether spikes inherently prevent transmission of subthreshold membrane potential fluctuations and shows that, in a high-dimensional setting, such information can be perfectly transmitted. Presynaptic neurons are modeled as N linear-nonlinear-Poisson units with a monotone transfer function; their subthreshold potentials are assumed to be a stationary Gaussian vector with a low-rank, approximately weakly correlated covariance generated from random Gaussian features; a postsynaptic population reads out membrane potentials by a dense linear decoder whose weights are proportional to the presynaptic covariance, enabling exact reconstruction in the large-N, large-P limit with P much smaller than N. The result holds for a broad class of transfer functions, including discontinuous ones, and relies on concentration of measure in high dimensions, implying that the nonlinear spiking nonlinearity becomes immaterial in this limit. Biologically, this predicts that subthreshold information can be recovered in desynchronized, high-dimensional cortical states, and suggests experimental tests using voltage imaging to compare reconstruction quality across brain states.

Abstract

Neurons primarily communicate through the emission of action potentials, or spikes. To generate a spike, a neuron's membrane potential must cross a defined threshold. Does this spiking mechanism inherently prevent neurons from transmitting their subthreshold membrane potential fluctuations to other neurons? We prove that, in theory, it does not. The subthreshold membrane potential fluctuations of a presynaptic population of spiking neurons can be perfectly transmitted to a downstream population of neurons. Mathematically, this surprising result is an example of concentration phenomenon in high dimensions.
Paper Structure (7 sections, 4 theorems, 55 equations, 1 figure)

This paper contains 7 sections, 4 theorems, 55 equations, 1 figure.

Table of Contents

  1. Acknowledgments.
  2. Proof of the theorem
  3. Proofs of the Lemmas
  4. Proof of Lemma \ref{['lemma:expectation']}
  5. Proof of Lemma \ref{['lemma:bounds']}
  6. Proof of Lemma \ref{['lemma:bias']}
  7. Generation of the membrane potentials in Fig. \ref{['fig:1']}

Key Result

Theorem 1

Let $\phi:\mathbb{R}\to\mathbb{R}_+$ be a non-constant, monotonically increasing function with at most polynomial growth, i.e., there exists an exponent $\alpha\geq 0$ such that $\limsup_{v\to+\infty}\frac{\phi(v)}{|v|^\alpha}<+\infty$. Then, if the membrane potential fluctuations satisfy the assump

Figures (1)

  • Figure 1: (Left panels) Presynaptic membrane potentials and spike trains of $N$ linear-nonlinear-Poisson neurons. The potentials $V_1(t), V_2(t), \dots, V_N(t)$ (blue lines). The instantaneous firing rate of a neuron depends nonlinearly on its potentials. Here, the nonlinear transfer function is the step function defined in \ref{['eq:threshold']}: a neuron emits stochastic spikes at rate $\rho$ only when the potential is above the threshold $\theta$ (dashed line). Taking $\theta = 1.65$, the potentials spend approximately $95\%$ of the time below the threshold (light blue areas). Taking $\rho = 20\,\text{Hz}$, the average firing rate of each neuron is approximately $1\,\text{Hz}$; as a result, spikes $S_1(t), S_2(t), \dots, S_N(t)$ (vertical bars) are sparse. At the single-neuron level, the spike train $S_i(t)$ contain little information about the potential $V_i(t)$. (Right panels) Comparison between the postsynaptic readouts and the presynaptic potentials. As predicted by the theorem, the readouts $\widehat{V}^{\Delta t}_1, \widehat{V}^{\Delta t}_2, \dots, \widehat{V}^{\Delta t}_N$ defined in \ref{['eq:readout']} (gray step-lines) give near-exact approximations of the true potentials $V_1(t), V_2(t), \dots, V_N(t)$ (blue lines, same as on the left panels) when $N$ and $P$ are large and $P\ll N$. The small deviations between the gray and blue traces are due to the fact that, while $N$ and $P$ are large, they are finite; here, $N=10^6$, $P=100$, and $\Delta t = 2\,\text{ms}$. The readouts are weighted sums of the spike trains: the weights are schematically represented by dotted lines and two weights, $W_{2,1}^{N,P}$ and $W_{1,N}^{N,P}$ are labeled as a example. At the network level, the spikes of a presynaptic population of neurons can perfectly transmit the presynaptic membrane potentials---including the subthreshold components (light blue area, left pannels)---to a postsynaptic population of linear readout neurons. The details of how the potentials $V_1(t), V_2(t), \dots, V_N(t)$ are generated in this figure are presented in Appendix \ref{['sec:details']}.

Theorems & Definitions (4)

  • Theorem
  • Lemma 1
  • Lemma 2
  • Lemma 3