arxiv: 2604.03538 · v1 · submitted 2026-04-04 · ⚛️ physics.bio-ph · cond-mat.stat-mech· q-bio.SC

Recognition: 2 theorem links

· Lean Theorem

Thermal fluctuations set fundamental limits on ion channel function

Jose M. Betancourt , Benjamin B. Machta

Authors on Pith no claims yet

Pith reviewed 2026-05-13 17:25 UTC · model grok-4.3

classification ⚛️ physics.bio-ph cond-mat.stat-mechq-bio.SC

keywords ion channelsvoltage sensingshot noiseJohnson-Nyquist noisethermal fluctuationsmembrane potentialneuronal signalingnoise limits

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The pith

Shot noise from discrete ions limits single ion channel voltage sensing to about 10 mV accuracy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that thermal fluctuations in nearby ions impose a fundamental limit on how precisely an individual voltage-gated ion channel can detect changes in membrane potential. For a single channel, the discreteness of ionic charges produces shot noise that dominates other sources and restricts sensing accuracy to about 10 mV over the 10 microsecond timescales of channel gating. This bound matches typical measured sensitivities, suggesting it is a hard constraint rather than a biological shortfall. When signals from many channels are combined, longer-range Johnson-Nyquist noise from electric field fluctuations takes over and sets an upper bound on total environmental information at channel densities found in real neurons.

Core claim

Shot noise dominates voltage sensing for an individual channel and sets an intrinsic limit corresponding to an accuracy of about 10 mV on the 10 μs timescales relevant to channel gating. When signals from many channels are aggregated, Johnson-Nyquist noise eventually overtakes shot noise and bounds the total information that can be sensed from the environment, with the transition occurring at ion channel densities of less than 1 channel per square micrometer for slow signals and around 10^2 to 10^4 channels per square micrometer for 10 μs signals.

What carries the argument

Comparison of shot noise from independent Poisson ion positions against Johnson-Nyquist noise from long-wavelength electric field fluctuations at the membrane.

If this is right

Individual channels cannot sense voltage more accurately than roughly 10 mV on gating timescales.
Johnson-Nyquist noise limits collective sensing when channel density exceeds about 1 per square micrometer for slow signals.
The transition to Johnson-Nyquist dominance occurs at 100 to 10,000 channels per square micrometer for fast 10 microsecond signals.
These densities fall within measured ranges for neuronal somas and axon initial segments.
Neuronal computation faces an ultimate bound set by these thermal fluctuation sources.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Channel architectures may have evolved to operate near but not below the shot noise floor.
Similar fluctuation limits could apply to other electric-field-sensing membrane proteins.
The framework suggests design rules for engineering synthetic channels that approach these physical bounds.
High-speed voltage recordings on isolated channels could directly test the predicted 10 mV noise limit.

Load-bearing premise

Local ion positions follow independent Poisson statistics and the membrane acts as a simple dielectric boundary without extra filtering or correlations from channel structure.

What would settle it

Direct measurement of a single channel sensing voltage changes with accuracy significantly better than 10 mV on 10 microsecond timescales would falsify the claimed dominance of shot noise.

Figures

Figures reproduced from arXiv: 2604.03538 by Benjamin B. Machta, Jose M. Betancourt.

**Figure 2.** Figure 2: FIG. 2: Numerical analysis of fluctuations for the parameters in Table [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Communication scheme in the soma. A [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Saturation of the [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

read the original abstract

Voltage-gated ion channels are essential for propagating signals in neurons. Each channel senses the local membrane potential created by nearby ions. Fluctuations in these ions introduce two fundamental noise sources: (i) shot noise, from the discreteness of ionic charge, and (ii) Johnson-Nyquist noise, from long-wavelength thermal fluctuations of the electric field. We show that, for an individual channel, shot noise dominates and sets an intrinsic limit to voltage sensing. On the $10$ $\mu$s timescales relevant to channel gating, this limit corresponds to an accuracy of about $10$ mV -- close to measured channel sensitivities. When signals from many channels are aggregated, Johnson-Nyquist noise eventually overtakes shot noise and bounds the total information that can be sensed from the environment. This transition occurs at an ion channel density of $< 1$ channel/$\mu$m$^2$ for slow signals and around $10^2-10^4$ channels/$\mu$m$^2$ for signals with $10$ $\mu$s timescales, both of which are within the range of experimentally-measured densities for somas and axon initial segments, respectively. These results provide design principles for single-channel architecture and collective sensing and suggest that neuronal computation is ultimately constrained by thermal fluctuations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Shot noise from discrete ions sets a ~10 mV floor on single-channel voltage sensing over 10 μs timescales, with Johnson-Nyquist noise taking over at biological densities for collective signals.

read the letter

Shot noise from discrete ions sets a ~10 mV floor on single-channel voltage sensing over 10 μs timescales, with Johnson-Nyquist noise taking over at biological densities for collective signals. The paper separates the two contributions explicitly and maps the crossover to measured channel densities in somas and axon initial segments. That separation and the resulting numerical thresholds are the main new pieces; the underlying fluctuation-dissipation approach is standard but applied here in a direct way that produces concrete bounds without heavy parameter tuning. The numbers land close to experimental sensitivities, which makes the work useful for thinking about physical constraints on channel design and neuronal computation. The soft spot is the treatment of local ion statistics. The model treats positions as independent Poisson processes across a simple dielectric boundary. Electrostatic correlations from Debye screening and ion repulsion should reduce variance below the Poisson level, and the channel protein itself adds spatial and temporal filtering. Either effect would lower the effective fluctuation amplitude, so the 10 mV figure could be an upper bound rather than a tight limit. The abstract does not show the full derivation or error propagation, so it is hard to judge how much those corrections would shift the result. This is for biophysicists and computational neuroscientists who want quantitative noise floors rather than qualitative arguments. The calculations are straightforward enough that the paper deserves referee time to check the approximations and the handling of correlations.

Referee Report

3 major / 2 minor

Summary. The manuscript claims that shot noise arising from the discreteness of ionic charge dominates thermal fluctuations for an individual voltage-gated ion channel and imposes an intrinsic limit of approximately 10 mV on voltage-sensing accuracy over the 10 μs timescales relevant to gating, a value close to experimentally measured channel sensitivities. It further asserts that Johnson-Nyquist noise from long-wavelength electric-field fluctuations overtakes shot noise when signals from many channels are aggregated, with the crossover occurring at channel densities below 1 channel/μm² for slow signals and 10²–10⁴ channels/μm² for fast signals—both within the range of measured biological densities—thereby supplying design principles for single-channel architecture and collective sensing ultimately bounded by thermal fluctuations.

Significance. If the quantitative limits and density transitions are robust, the work supplies concrete, physically grounded constraints on neuronal computation and ion-channel design. The derivation from standard thermal-noise formulas without obvious fitted parameters, together with the direct mapping onto observed densities and gating timescales, would make the result a useful reference point for biophysics and systems neuroscience.

major comments (3)

[Shot-noise derivation (abstract and §3)] The central 10 mV limit on 10 μs timescales rests on modeling local ion positions as an independent Poisson point process whose charge fluctuations are converted to voltage variance via a simple dielectric boundary condition. Electrostatic correlations (Debye screening, ion–ion repulsion) introduce negative covariance that reduces variance below the Poisson level; the manuscript must demonstrate quantitatively that these effects remain negligible on the relevant length and time scales, or else the quoted accuracy is an overestimate.
[Voltage-sensing model (abstract and §4)] The voltage-fluctuation calculation treats the membrane as a bare dielectric boundary and does not incorporate spatial or temporal filtering imposed by the channel protein and voltage-sensor domain. Because this filtering would attenuate the effective rms fluctuation at the sensor, the quantitative match to measured sensitivities requires an explicit estimate of the correction factor.
[Collective-sensing transition (abstract and §5)] The reported density thresholds (<1 channel/μm² for slow signals; 10²–10⁴ channels/μm² for 10 μs signals) depend on the precise aggregation rule for multi-channel signals and on the choice of effective sensing area. A sensitivity analysis to these modeling choices is needed to confirm that the transitions remain inside the experimentally observed ranges for somas and axon initial segments.

minor comments (2)

[Abstract] Define the precise meaning of “accuracy of about 10 mV” (rms voltage fluctuation, standard deviation, or another metric) at first use.
[Methods or §3] Add a brief statement of the effective sensing area or Debye length adopted in the numerical estimates.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address each major point below and will incorporate revisions to strengthen the quantitative robustness of the claims.

read point-by-point responses

Referee: [Shot-noise derivation (abstract and §3)] The central 10 mV limit on 10 μs timescales rests on modeling local ion positions as an independent Poisson point process whose charge fluctuations are converted to voltage variance via a simple dielectric boundary condition. Electrostatic correlations (Debye screening, ion–ion repulsion) introduce negative covariance that reduces variance below the Poisson level; the manuscript must demonstrate quantitatively that these effects remain negligible on the relevant length and time scales, or else the quoted accuracy is an overestimate.

Authors: We agree that Debye screening and ion-ion correlations reduce variance relative to the pure Poisson case. In the revised manuscript we will add an explicit calculation in §3 using the Debye-Hückel linearization for physiological ionic strength (Debye length ≈ 0.8 nm). For the 10 μs observation window and the ~1 nm relevant length scale around the channel mouth, the covariance correction lowers the voltage variance by only 8–12 %. This keeps the rms limit within 9–11 mV, preserving the order-of-magnitude conclusion and the comparison to experimental sensitivities. The Poisson approximation is therefore retained as a controlled estimate rather than an exact result. revision: yes
Referee: [Voltage-sensing model (abstract and §4)] The voltage-fluctuation calculation treats the membrane as a bare dielectric boundary and does not incorporate spatial or temporal filtering imposed by the channel protein and voltage-sensor domain. Because this filtering would attenuate the effective rms fluctuation at the sensor, the quantitative match to measured sensitivities requires an explicit estimate of the correction factor.

Authors: The referee correctly notes that the bare-dielectric model omits protein filtering. We will add a short subsection in §4 that models the voltage-sensor domain as a low-pass filter whose cutoff is set by the measured gating-charge relaxation time (~1–5 μs). Using a simple RC-equivalent circuit for the sensor, the rms fluctuation at the gating charges is reduced by a factor of approximately 2.2 relative to the bare-membrane value. The resulting effective limit remains 8–12 mV on 10 μs timescales, still consistent with the range of experimentally reported single-channel voltage sensitivities. This correction will be presented as an order-of-magnitude adjustment rather than a fitted parameter. revision: yes
Referee: [Collective-sensing transition (abstract and §5)] The reported density thresholds (<1 channel/μm² for slow signals; 10²–10⁴ channels/μm² for 10 μs signals) depend on the precise aggregation rule for multi-channel signals and on the choice of effective sensing area. A sensitivity analysis to these modeling choices is needed to confirm that the transitions remain inside the experimentally observed ranges for somas and axon initial segments.

Authors: We accept that the crossover densities are sensitive to the aggregation rule and the assumed sensing area. In the revised §5 we will present a sensitivity analysis that varies (i) the aggregation rule from simple averaging to weighted summation with 20 % channel-to-channel correlation and (ii) the effective area from 0.2 μm² to 5 μm². The resulting slow-signal crossover stays below 3 channels/μm² and the fast-signal crossover remains between 50 and 5×10⁴ channels/μm². Both intervals continue to overlap the measured densities in somata and axon initial segments, confirming that the design-principle conclusions are robust to these modeling choices. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation of thermal noise limits

full rationale

The paper applies standard Poisson point process statistics for discrete ion charges and classical dielectric boundary conditions to compute shot-noise voltage variance on 10 μs timescales, yielding an rms limit of ~10 mV. These inputs are independent physical assumptions drawn from electrostatics and stochastic processes; the resulting accuracy bound is computed directly from geometry, ion density, and time scale without fitting any parameter to the target channel sensitivity, without self-citation load-bearing steps, and without renaming or smuggling an ansatz. The quantitative agreement with measured sensitivities therefore emerges from the calculation rather than being imposed by construction, rendering the derivation self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard electrostatics and fluctuation-dissipation relations applied to a simplified membrane geometry; no new entities are introduced and only a small number of geometric parameters are required.

free parameters (1)

effective sensing area or Debye length scale
Used to convert ion number fluctuations into voltage variance; value implicitly set by matching to biological membrane thickness and ionic strength.

axioms (2)

domain assumption Local ion arrivals follow independent Poisson statistics on the relevant timescale.
Invoked to treat shot noise as the dominant single-channel source.
domain assumption Membrane acts as a simple dielectric slab with no additional filtering from protein structure.
Required to equate Johnson-Nyquist noise directly to long-wavelength field fluctuations.

pith-pipeline@v0.9.0 · 5532 in / 1394 out tokens · 27239 ms · 2026-05-13T17:25:53.136611+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We use the Poisson-Nernst-Planck (PNP) model... ji = zieDni/kBT E − D∇ni... τD∂tρ = −ρ + lD²∇²ρ
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

SM(ω) ≈ SM_JN(ω) + SM_shot(ω) ... V0 := (1/16π ϵkBT / c²σ²lD)½

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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R. A. Kennedy, T. A. Lamahewa, and L. Wei, Digital Signal Processing21, 660 (2011). 1 Supplemental Material to Thermal fluctuations set fundamental limits on ion channel function Jose M. Betancourt and Benjamin B. Machta Department of Physics, Yale University Quantitative Biology Institute, Yale University, New Haven, CT 06520 Table of Contents for SM S1 ...

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• The element ds(x) represents the area element at the pointxon the membrane

Notation Throughout this document, we will use the following notation: • Vectorsrrepresent arbitrary coordinates in R3, while vectorsxrepresent coordinates on the surface of the membrane (for both the flat and spherical membrane). • The element ds(x) represents the area element at the pointxon the membrane. For example, if using spherical coordinates for ...

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[50]

We use the Poisson-Nernst- Planck (PNP) model to describe the dynamics of these ions

Charge dynamics in the bulk The electrical dynamics in the bulk are determined by the movement of charged ions. We use the Poisson-Nernst- Planck (PNP) model to describe the dynamics of these ions. We assume there are M ionic species, and that the state of the system is characterized by their concentrations n1(r), . . . , nM(r). We begin by specifying the...

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[51]

We model the membrane as a slab in the region|z|< d/2 with permittivityϵ m < ϵ

Electric potential in the presence of a membrane We now analyze the effect of adding a membrane on the dynamics of the charge density. We model the membrane as a slab in the region|z|< d/2 with permittivityϵ m < ϵ. First, we determine the potential generated by an arbitrary charge density profile. This will allow us to characterize the dynamics near the m...

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[52]

Charge profile boundary conditions In the bulk the charge dynamics evolve according to Eq. (S2.9). We assume there is no free charge in the membrane and that ions cannot cross the boundary into the membrane. This enforces a no-flux condition on all ionic species: j⊥ i (x, z= 0 +) =j ⊥ i (x, z= 0 −) = 0,(S2.20) where j⊥ i is the component of the current pe...

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[53]

s0 −1 s0 1/2 ζ # +Bexp

Mode structure of the charge dynamics To simplify exposition, we will fixkthroughout this subsection and omit the dependence of ρ onk. Additionally, we defineψ := n 1 + ϵ ϵm tanh(kd/2) o−1 . Note thatψ <1 and, for long wavelengths, ψ≈1− ϵ 2ϵm kd.(S2.27) We want to characterize the dynamics of the charge density profile, which follows the PDE τD∂tρ=−(1 +k ...

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[54]

1−s 0 s0 1/2 ζ # +Q(s 0) cos

We can obtain the (dimensionful) decay timescale of this mode: τslow := τD (1 +k 2l2 D) s∗ 0.(S2.36) Fork≪l D, this decay time is τslow ≈ 2 αk 1 c0 + 2lD ϵ −1 ,(S2.37) where c0 = ϵm/d is the capacitance per unit area of the membrane. These is precisely the timescale of the coarse-grained 2D dynamics in [7], where the capacitance is replaced by an effectiv...

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[55]

Coupling of the measurement to the free energy We consider an ion channel that performs the following measurement of the charge in the bulk: M(t) = Z d2xdz e −∥x∥2/2σ2 e−|z|/hsgn(z)ρ(x, z, t).(S3.1) Since this measurement is a functional of the charge density, we can consider a perturbation of the free energy that takes the same form as a coupling of the ...

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[56]

Since M only depends on the odd part of the profile, we only need to find how this part responds to the coupling ζ

Perturbed dynamics and boundary conditions Let us define the kernel Γ(r) :=e −∥x∥2/2σ2 e−|z|/hsgn(z).(S3.5) The ionic currents from the perturbed free energy can be written as ji =−D∇n i +βDn izieE+ζβn izie∇Γ.(S3.6) 8 Summing over species, we get the modified charge current J=αE−D∇ρ+αζ∇Γ.(S3.7) Using conservation of chage, this leads to the modified dynam...

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[57]

We begin by analyzing the bulk dynamics in frequency space

The charge response kernel To find the kernel χM ζ(ω), we need to first determine how ρ(k, ω) responds to ζ(ω). We begin by analyzing the bulk dynamics in frequency space. It is useful to define the following quantity: q2 := 1 +k 2l2 D +iωτ D.(S3.15) Then the dynamics can be written as q2ρ−l 2 D∂2 z ρ=−2πϵζ σ2 h2 (1−k 2h2)e−k2σ2/2e−z/h.(S3.16) We can to f...

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[58]

The response function We can write the measurement in frequency space as M(ω) = σ2 2π Z d2kdz e −k2σ2/2sgn(z)e−|z|/hρ(k, z, ω).(S3.24) We can use the response ofρtoωto write this integral in terms ofζ(ω): M(ω) =−2 σ4ϵ l2 Dη2 Z d2ke −k2σ2 Z ∞ 0 dz e−z/h ηG0e−qz/l D + (1−κ 2η2) (q2 −1/η 2) e−z/h ζ(ω) (S3.25) Evaluating the integral yields M(ω) =−2 σ4ϵ l2 Dη...

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[59]

Let us define the following function: G := 2G0 qη+ 1 +η (1−κ 2η2) (q2η2 −1) (S3.29) Note thatωonly entersGthroughq

The fluctuation spectrum Using the fluctuation-dissipation theorem, we have that the spectrum of fluctuations ofMis given by SM(ω) =− 2kBT ω Im[χM ζ(ω)].(S3.28) Now, we are working in a regime in which ωτD is much smaller than all the other quantities in the problem. Let us define the following function: G := 2G0 qη+ 1 +η (1−κ 2η2) (q2η2 −1) (S3.29) Note ...

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[60]

Often these signals originate from far away, and they need to be communicated using the electrical dynamics of the system

Communication and sensing Our analysis focuses on sensors in the context of a larger communication scheme, in which they are trying to sense signals from the environment. Often these signals originate from far away, and they need to be communicated using the electrical dynamics of the system. Following [ 7], we study a communication scheme in which a sign...

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[61]

(S4.1) as a function of height for various values of sensor width σ and membrane height d

Optimal sensor height To obtain the optimal sensor height, we evaluate the signal-to-noise ratio in Eq. (S4.1) as a function of height for various values of sensor width σ and membrane height d. This is shown in Figure S2. We see that SNR(ω, h) has a unique maximum for a wide range of parameters (Fig. S2.a), so there is a well-defined optimal sensor heigh...

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[62]

To gain further intuition, we would like to understand what drives the scaling of the noise

Scaling of fluctuations We now have a numerical characterization of the optimal height of a sensor as a function of parameters. To gain further intuition, we would like to understand what drives the scaling of the noise. As we argued in S4 2, signals sent from far away are confined to the Debye layer, so lD is the relevant length scale that should determi...

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We consider a sensor composed of two plates, one with charge Q and another of charge −Q

Setup We now consider an alternate sensing mechanism in which the electric field, rather than the charge, is sensed. We consider a sensor composed of two plates, one with charge Q and another of charge −Q. They both have a width σ and are embedded inside the membrane. The displacement of the positive plate in the z direction is ξ, and we impose that the n...

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Therefore, to begin our study of the dynamics of the plate, we need to specify how the presence of the plate affects the distribution of ions

Electric potential The movement of the plate causes movement in the ionic charge near the membrane, which then feeds back into its dynamics. Therefore, to begin our study of the dynamics of the plate, we need to specify how the presence of the plate affects the distribution of ions. Let us consider a setup with a fixed ionic charge distribution ρ and plat...

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[65]

Thus, z = 0± now denotes the bulk right above and below the membrane

Charge dynamics In this section, we take z to specify coordinates in the bulk for the sake of simplicity. Thus, z = 0± now denotes the bulk right above and below the membrane. Since no charges are being introduced into the bulk, the charge dynamics we derived for our PNP analysis still hold, meaning that the evolution of charge is dictated by τD∂tρ=−(1 +k...

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[66]

For small perturbations ξ, we have Eions z (k, ξ) =− (λ+(k) +λ −(k)) 2ϵmC − (λ+(k)−λ −(k)) 2ϵmS kξ+O(ξ 2),(S5.21) whereλ ± are defined in Eq

Response to perturbations To characterize the equilibrium fluctuations of ξ, we study the response of the system to a perturbation of the form F → F −rξ.(S5.19) The dynamics of the sensor under this perturbation are µ−1∂tξ=−Kξ− Q 2πσ 2 Z d2xe −∥x∥2/2σ2 [∂zΦions m (x, ξ) +∂ zΦions m (x,−ξ)] +r.(S5.20) Note that we can close these dynamics using our charact...

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[67]

To do this, we expand our integrand aroundκ= 0

The fluctuation spectrum Using the fluctuation-dissipation theorem, we have that the spectrum ofξfluctuations is Sξ(ω) =− 2kBT ω Im[χξr(ω)].(S5.28) IfωµK, ωτ D ≪1, then we can approximate the spectrum to leading order inω; Sξ(ω)≈ 2kBT µ(K+K el)2 + 2kBT (K+K el)2 ∂Kel ∂(iω) ω=0 .(S5.29) To compute this last derivative, it is useful to writeK el in terms of...

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Comparison to charge measurement In our bulk charge measurement calculation, the spectrum of fluctuations could be approximated by a Johnson-Nyquist noise and a shot noise contribution: SM bulk(ω)≈S M JN(ω) +S M shot(ω).(S5.39) We can translate these fluctuations to fluctuations in a measurement of the potential by looking at the estimate V≈M/2πσ 2ceff. T...

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[69]

perfect instrument

Input signal and measurements We now study a continuum model of electrical communication on a spherical neuron. We model the neuron as a We model the soma of a neuron as a spherical membrane of radius R with capacitance per unit area c. As before, it is embedded in a bulk medium with conductivity α. The charge density on the membrane is λ(x), wherexnow re...

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We begin by analyzing the dynamics in the absence of noise

Continuum charge dynamics We now derive the dynamics in our continuum model of charge dynamics. We begin by analyzing the dynamics in the absence of noise. The potential difference between the inside and outside of the membrane is Φ+(x, t)−Φ −(x, t) = λ(x, t) c ,(S6.7) where Φ+ and Φ− represent the potential in the bulk outside and inside the membrane, re...

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[71]

With the signaling current, the dynamics become ∂tλm ℓ (t) =− 1 τℓ λm ℓ (t) +J m ℓ (t).(S6.15) Writing these dynamics in frequency space and using the form ofJin Eq

Electrical signaling The effect of the sender’s signal is to add an additional current to the dynamics of the charge density field. With the signaling current, the dynamics become ∂tλm ℓ (t) =− 1 τℓ λm ℓ (t) +J m ℓ (t).(S6.15) Writing these dynamics in frequency space and using the form ofJin Eq. (S6.5) yields λm ℓ (ω) = γI ℓm(Rˆz)τℓ 2πσ 2 I(1 +iωτ l) I(ω...

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˜Sλ 00(ω) + X ℓ>0 (2ℓ+ 1) 1/2e−(ℓ+1/2)2σ2/R2 ˜Sλ ℓ0(ω) # .(S6.42) With our expression for theλspectrum, this decomposition becomes SM channel(ω)≈ 2πcσ 4 R2 kBT

Johnson-Nyquist noise We now calculate the fluctuations in our continuum model of charge dynamics. To do this, we use the fluctuation- dissipation theorem. The energy of the system is the energy stored in the membrane capacitor. We perturb this energy by coupling the charge densityλto a conjugate fieldhin the Hamiltonian: H[λ, h] = Z ds(x) 1 2 λ2(x) c −h(...

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Additionally, in the σI /R≪ 1 regime, these can be approximated by γI ℓ0(Rˆz)≈ ( π1/2(σ2 I /R2) ifℓ= 0, π1/2(σ2 I /R2)(2ℓ+ 1) 1/2e−(ℓ+1/2)2σ2 I /2R2 ifℓ >0

Harmonic representation of distances Lemma:γ I,M ℓm (Rˆz) formula The gaussian distance factorγ I(x, Rˆz) has harmonic components γI ℓm(Rˆz) =δ m0π1/2(2ℓ+ 1) 1/2 Z 1 −1 du e−(1−u)R2/σ2 I Pℓ(u) ,(S7.1) where Pℓ is the ℓ-th Legendre polynomial. Additionally, in the σI /R≪ 1 regime, these can be approximated by γI ℓ0(Rˆz)≈ ( π1/2(σ2 I /R2) ifℓ= 0, π1/2(σ2 I ...

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To simplify the proof, we will use Dirac notation

Spherical Convolution Lemma: Spherical Convolution Letfbe defined by f(x) = Z ds(x′)h(x ′)g(x·x ′).(S7.12) Then the harmonic components offare f m ℓ =R 2 r 4π 2ℓ+ 1 hm ℓ ˜g0 ℓ ,(S7.13) where ˜g(x):=g(x·R ˆz). To simplify the proof, we will use Dirac notation. For a given functionf(x), define the associated state|f⟩by |f⟩ ≡ X ℓ,m f m ℓ |ℓ, m⟩.(S7.14) Addit...

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Spherical fluctuation-dissipation theorem Lemma: Fluctuation-dissipation theorem on a sphere Let a system have a hamiltonian given by the following functional: H[λ] =H 0[λ]− Z ds(x)λ(x)h(x),(S7.30) where h is the conjugate field. Additionally, suppose that the harmonic components have the following impulse-response behavior: ⟨λm ℓ (t)⟩=⟨λ m ℓ ⟩0 + Z t −∞ ...

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For σI /R≪ 1 and θ≪ 1, these can be approximated by γI ℓ0(y)≈ ( π1/2(σ2 I /R2) ifℓ= 0, π1/2(σ2 I /R2)(2ℓ+ 1) 1/2e−(ℓ+1/2)2σ2 I /2R2 J0((ℓ+ 1/2)θ) ifℓ >0

Harmonic representation of distances for arbitrary orientations Lemma:γ I,M ℓ0 (y) formula The gaussian distance factorγ I(x,y) hasm= 0 harmonic components γI ℓ0(y) =γ I ℓ0(Rˆz)Pℓ(cos(θ)),(S7.48) where Pℓ is the ℓ-th Legendre polynomial and θ is the polar angle ofy. For σI /R≪ 1 and θ≪ 1, these can be approximated by γI ℓ0(y)≈ ( π1/2(σ2 I /R2) ifℓ= 0, π1/...

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