White Matter Capacity

Introduction

The axon tract, a bundle of parallel axons, is traditionally studied in neurobiology as a medium for signal propagation. Recent theoretical work has revealed that such tracts can be viewed through the lens of information theory as communication channels. In particular, ephaptic coupling—the interaction between axons mediated by extracellular fields—introduces cross-talk that resembles multiple-input multiple-output (MIMO) systems.

Chapter 5 of the dissertation [1] introduced two modifications to the generalized ephaptic equation: (i) a position-dependent $W_{ip}(z)$ matrix capturing tomographic geometry, and (ii) an additive Gaussian noise term modeling ionic thermal fluctuations. Together, these modifications yield a tract model that behaves like a vector Gaussian channel with perfect feedback. The present paper develops the explicit capacity analysis of this channel.

Table 1: Table of Notation summarizes the principal symbols and meanings as used in this work.

Background: Ephaptic Equation and Modifications

The generalized ephaptic equation governs the dynamics of transmembrane voltages $V_i(z,t)$ along axons. Two key modifications were introduced:

Modification A: Position-dependent $W_{ip}(z)$

Tomographic data reveals complex geometrical arrangements of axons. To incorporate this, the interaxonal coupling matrix $W_{ip}$ is made position-dependent: \[ W_{ip} \;\; W_{ip}(z), \] yielding a differentiable matrix function along the tract axis. This allows spatially varying coupling strengths.

Modification B: Additive Gaussian Noise

The extracellular medium contains ions undergoing random thermal motion. This is modeled as additive Gaussian noise $n_{ip}(z,t)$ added to ephaptic signals: \[ \frac{\partial^2 V_i}{\partial z^2} \mapsto \frac{\partial^2 V_i}{\partial z^2} + n_{ip}(z,t). \] This transforms the tract into a noisy communication channel.

Tract as a Vector Gaussian Channel

Discretizing the ephaptic equation via the Crank–Nicolson scheme yields:

where $A$ and $B$ are coefficient matrices, $V$ is the present voltage vector, $\mathcal{V}$ the next-step voltage vector, and $j_{{ion}}, j_{{stim}}$ are ionic and stimulation currents.

Inverting $A$ via singular value decomposition gives:

with $K = A^{-1}B$, $R$ and $S$ matrices coupling ionic variables, and $T_2$ nonlinear terms. This system can be expressed as:

where $Y_k$ is the output state, $X_{k+1}$ the input stimulus, $D(\cdot)$ deterministic feedback, and $N_{k+1}$ Gaussian noise. Thus, the tract acts as a vector Gaussian channel with perfect feedback.

Feedback Elimination

Because $D(\cdot)$ is deterministic and causal, it can be eliminated by algebraic manipulation. Stacking outputs over $n$ epochs: \[ \mathbf{Y} = \mathbf{D}\mathbf{Y} + \mathbf{G}\mathbf{X} + \mathbf{N}. \] Solving, \[ \mathbf{Y} = (\mathbf{I}-\mathbf{D})^{-1}\mathbf{G}\mathbf{X} + (\mathbf{I}-\mathbf{D})^{-1}\mathbf{N}. \] Define effective channel and noise: \[ \mathbf{H}_{\text{eff}} = (\mathbf{I}-\mathbf{D})^{-1}\mathbf{G}, \Sigma_{\text{eff}} = (\mathbf{I}-\mathbf{D})^{-1}\Sigma_N((\mathbf{I}-\mathbf{D})^{-1})^T. \] The tract reduces to: \[ \mathbf{Y} = \mathbf{H}_{\text{eff}}\mathbf{X} + \mathbf{Z}, \quad \mathbf{Z} \sim \mathcal{N}(0,\Sigma_{\text{eff}}). \]

Finite-block Capacity

For block length $n$, input covariance $\mathbf{K}_X$, and power constraint $\frac{1}{n}\text{tr}(\mathbf{K}_X) \le P$, the capacity is:

Maximization over $\mathbf{K}_X$ yields the optimal allocation.

Waterfilling

Diagonalizing \[ R = \Sigma_{\text{eff}}^{-1/2}\mathbf{H}_{\text{eff}}\mathbf{H}_{\text{eff}}^T\Sigma_{\text{eff}}^{-1/2}, \] reveals eigenmodes. Power is allocated across these modes via waterfilling: \[ p_i = (\mu - \frac{1}{\lambda_i})^+, \] where $\lambda_i$ are eigenvalues of $R$, $\mu$ chosen to satisfy $\sum p_i = P$.

Stationary Infinite-horizon Capacity

If $\mathbf{H}_{\text{eff}}$ and $\Sigma_{\text{eff}}$ are block Toeplitz (stationary), as $n$ capacity per epoch converges to:

where $S_Z(\omega)$ is noise spectral density, $S_{YX}(\omega)$ channel frequency response, and $S_X(\omega)$ input spectrum optimized under power constraint.

Illustrative Examples

Without Ephaptic Coupling

Setting $W_{ip}(z)=0$ decouples axons. $\mathbf{H}_{\text{eff}}$ becomes block-diagonal, yielding independent channels. Capacity is sum of individual axon capacities.

With Ephaptic Coupling

Nonzero $W_{ip}(z)$ introduces cross-terms, enlarging $\mathbf{H}_{\text{eff}}$. Eigenvalues of $R$ increase, allowing higher capacity via spatial multiplexing. Thus, ephaptic coupling can enhance tract capacity.

Discussion

The tract capacity analysis reveals several insights:

• Feedback neutrality: Deterministic feedback does not reduce capacity; it is absorbed into effective channel parameters.

• Noise shaping: Position-dependent $W_{ip}(z)$ and colored noise $\Sigma_{\text{eff}}$ create nontrivial spectral dependencies.

• Biophysical constraints: Power constraint $P$ corresponds to stimulation energy limits; realistic values must be estimated from neurophysiology.

• Ephaptic gain: Coupling can increase capacity, but also introduces correlated noise; net effect depends on geometry and noise statistics.

Capacity per Unit Cost: Verdu Regime

Motivation

In biological communication systems, energetic efficiency rather than average power is often the fundamental constraint. Ionic pumping, membrane recovery, and stimulation currents incur a metabolic cost that scales with signal amplitude and duration. This motivates analyzing the capacity per unit cost, as introduced by Verdu, which characterizes the maximum achievable information rate per unit energetic expenditure in the low-cost regime.

This framework is particularly relevant for axon tracts, where signaling is often sparse and near-resting, and where efficiency constraints may dominate over raw throughput.

Effective Channel Model

From Section 4, the axon tract reduces to an equivalent memoryless vector Gaussian channel with deterministic feedback eliminated:

where $Z \sim \mathcal{N}(0,\Sigma_{\text{eff}})$ and $X \in \mathbb{R}^m$ denotes the stimulation input vector.

We assume the existence of a zero-cost input symbol $X=0$, corresponding to the resting state of the tract.

Cost Function

We associate a quadratic cost function to each input:

where $Q \succ 0$ is a symmetric positive-definite matrix modeling energetic or metabolic cost. The choice $Q=I$ corresponds to isotropic energy expenditure, while diagonal $Q$ allows axon-dependent energetic efficiencies.

Verdu Capacity per Unit Cost

For a memoryless channel with a zero-cost input, Verdu showed that the capacity per unit cost is given by

where $D(\cdot\|\cdot)$ denotes the Kullback–Leibler divergence.

This expression characterizes the slope of the capacity–cost curve at zero cost and avoids explicit blocklength optimization.

Evaluation for the Axon Tract

For the effective Gaussian channel,

The KL divergence evaluates to

Substituting into Verdu's formula yields

Generalized Eigenvalue Characterization

The supremum is achieved by the principal generalized eigenvector of the matrix pair

Hence, the capacity per unit cost admits the closed-form expression

in nats per unit cost. Division by $ 2$ converts this quantity to bits per unit cost.

Relation to Power-Constrained Capacity

Let $C(P)$ denote the power-constrained capacity derived in Sections 5 and 6. Then

showing that the Verdu regime corresponds to the low-power limit of the classical capacity expression.

Implications for Axon Tracts

The capacity per unit cost provides a sharp efficiency bound for tract-level communication:

• Optimal signaling at vanishing energy aligns with the most efficient eigenmode of the ephaptically coupled tract.

• Ephaptic coupling can increase information efficiency even when total capacity gains are modest.

• Deterministic feedback remains capacity-neutral in the low-cost regime.

• The framework directly links information flow to metabolic constraints, offering a principled notion of bits per unit biological cost.

Relation to Berger–Levy Energy-Efficient Neural Communication

Overview

Berger and Levy [2] developed a seminal information-theoretic theory of neural communication focused on single-neuron spike timing codes optimized for bits per joule. Their framework models a neuron as a stochastic encoder that converts afferent excitation into interspike intervals (ISIs), with optimization performed over ISI distributions under explicit metabolic cost constraints.

The present work addresses a complementary but fundamentally distinct problem: the collective communication capacity of an axon tract, modeled as a vector Gaussian channel with ephaptic coupling and perfect feedback. This section clarifies the relationship between the two formulations and delineates their respective domains of validity.

Level of Description

A primary distinction lies in the level at which communication is modeled.

• Berger–Levy: A single neuron is treated as the channel. Information is encoded in the timing of spikes, specifically in the distribution of interspike intervals generated by an integrate-and-fire–type mechanism driven by Poisson afferent bombardment.

• Present work: An axon tract is treated as the channel. Information is encoded in continuous-valued transmembrane voltage vectors evolving along space and time, with ephaptic coupling inducing MIMO-like interactions.

Thus, Berger–Levy operate at the level of discrete spike events and their timing statistics, whereas the present formulation operates at the level of spatially extended, analog biophysical fields prior to spike generation.

Channel Models

The underlying channel models differ both structurally and mathematically.

agraph{Berger–Levy} The neuron is modeled as a timing channel whose output is a point process. Noise is primarily Poissonian and arises from stochastic afferent spike arrivals. The channel is nonlinear and history-dependent, with the ISI distribution emerging endogenously from the dynamics.

agraph{Present Work} After discretization and feedback elimination, the axon tract reduces to an effective vector Gaussian channel \[ \mathbf{Y} = \mathbf{H}_{\text{eff}}\mathbf{X} + \mathbf{Z}, \] with colored Gaussian noise and deterministic feedback absorbed into $\mathbf{H}_{\text{eff}}$ and $\Sigma_{\text{eff}}$. This enables the use of classical tools such as eigenmode decomposition, waterfilling, and spectral capacity formulas.

In short, Berger–Levy emphasize Poisson timing noise, whereas the tract model emphasizes Gaussian field fluctuations and coupling.

Role of Energy and Cost

Both frameworks are deeply concerned with energetic efficiency, but they formalize cost differently.

• Berger–Levy: Energy is expended per ISI and per spike. Optimization explicitly trades bits per second against bits per joule, leading to gamma-distributed ISIs and beta-distributed excitation statistics.

• Present work: Energy enters via a quadratic cost function on continuous-valued inputs, \[ c(x) = x^T Q x, \] and efficiency is quantified via Verdu’s capacity per unit cost. The result is governed by a generalized eigenvalue problem involving $\mathbf{H}_{\text{eff}}$ and $\Sigma_{\text{eff}}$.

The Berger–Levy cost is event-based and metabolic, while the present cost is signal-based and field-theoretic, reflecting different dominant biophysical expenditures at different scales.

Feedback and Memory

Feedback plays markedly different roles in the two theories.

Berger–Levy explicitly account for history dependence arising from refractoriness, excitation correlation time, and ISI-to-ISI dependence, leading to information-rate decrements due to temporal correlations.

In contrast, the present work shows that deterministic, causal feedback in the tract dynamics does not reduce capacity: it can be eliminated algebraically and absorbed into an effective channel. This difference reflects the fact that Berger–Levy feedback is stochastic and energetic, whereas tract-level feedback is deterministic and biophysical.

Complementarity of the Two Approaches

The two formulations should be viewed as complementary rather than competing.

• Berger–Levy characterize how a single neuron optimally converts analog excitation into sparse spike timing under metabolic constraints.

• The present work characterizes how a population of axons collectively transports analog information prior to discretization into spikes.

Conceptually, the tract capacity analyzed here provides an upper bound on the information available to downstream spike encoders of the Berger–Levy type. Conversely, Berger–Levy’s bpj-optimal spike codes can be interpreted as candidate mechanisms for approaching this bound at the single-neuron level.

Summary of Key Differences

Table II: Key Differences summarize the key points of variation between the Berger-Levy and present approaches.

Perspective

Taken together, Berger–Levy and the present work suggest a multiscale picture of neural communication: energy-efficient spike timing emerges as an optimal encoding layer built atop a tract-level analog channel whose capacity and efficiency are governed by geometry, coupling, and noise. Bridging these two levels remains an open and promising direction for future work.

Relation to Ikeda–Manton Capacity Formulations

Overview

Ikeda and Manton developed an information-theoretic framework for the capacity of continuous-time channels governed by stochastic differential equations (SDEs), with particular emphasis on nonlinear dynamics and additive Gaussian diffusion noise. Their work characterizes mutual information rates for channels of the form

where $f()$ is generally nonlinear, $X_t$ is the control or input process, and $W_t$ is a Wiener process. Capacity is defined as a supremum over admissible input processes subject to energy or power constraints.

The present work addresses a related but distinct problem: the information-carrying capacity of an axon tract governed by biophysical field equations with ephaptic coupling, discretized in space and time and subsequently reduced to an effective vector Gaussian channel.

Modeling Philosophy

Ikeda and Manton treat the channel as an irreducibly nonlinear stochastic dynamical system. The signal and noise are intertwined at the level of the diffusion process, and capacity is defined directly on the path space of trajectories. Feedback enters naturally through the state dependence of the drift term $f(Y_t,X_t)$.

In contrast, the present work begins from a nonlinear biophysical model (the generalized ephaptic equation) but demonstrates that, after discretization and algebraic manipulation, the deterministic nonlinear feedback can be eliminated. This yields an equivalent linear vector Gaussian channel with modified channel matrix $\mathbf{H}_{\text{eff}}$ and noise covariance $\Sigma_{\text{eff}}$. Capacity analysis is then performed using classical Gaussian tools.

Thus, while Ikeda–Manton retain the nonlinear SDE structure throughout, the present work shows that for axon tracts the relevant nonlinearities are capacity-neutral and can be absorbed into effective linear parameters.

Treatment of Feedback

Feedback plays a central role in both formulations, but with different consequences.

In Ikeda–Manton, feedback is intrinsic to the continuous-time dynamics and generally cannot be removed without altering the channel law. Capacity expressions involve innovations processes and Radon–Nikodym derivatives on path space, reflecting the deep coupling between signal, state, and noise.

In the present work, feedback arising from ephaptic interactions and ionic dynamics is deterministic and causal. As shown in Section 4, this feedback can be eliminated exactly, leaving capacity unchanged. This distinction highlights a key structural difference: Ikeda–Manton feedback is stochastic and diffusion-level, whereas tract-level feedback is deterministic and field-mediated.

Noise Structure

Ikeda and Manton model noise as continuous-time Brownian motion acting directly on the system state, leading to nontrivial stochastic calculus considerations and capacity expressions involving quadratic variation.

By contrast, the axon tract model yields additive Gaussian noise after discretization, with spatial and temporal correlations captured in $\Sigma_{\text{eff}}$. This permits an eigenmode-based interpretation of capacity and an explicit connection to MIMO Gaussian channels.

Capacity Notions

Capacity in Ikeda–Manton is defined as an information rate per unit time over continuous trajectories, typically under average power constraints on $X_t$. Closed-form expressions are rare and generally limited to special cases.

The present work derives explicit finite-block, stationary, and capacity-per-unit-cost expressions. In particular, the Verdu regime yields a closed-form eigenvalue characterization, which has no direct analog in the general Ikeda–Manton setting.

Conceptual Relationship

The present formulation can be viewed as a biophysically grounded specialization of the general Ikeda–Manton framework to a class of systems where nonlinear feedback is deterministic and noise enters additively. From this perspective, the tract model identifies a regime in which the formidable path-space capacity problem collapses to a tractable Gaussian channel without loss of rigor.

Summary

Ikeda and Manton provide a general and mathematically sophisticated theory of capacity for nonlinear stochastic dynamical channels. The present work complements this theory by showing that axon tracts occupy a structurally simpler subclass in which effective linearization is exact, enabling explicit capacity and efficiency results directly tied to neurobiological structure.

Multivariate Mutual Information of the Axon Tract

Vector-Valued Channel Representation

After feedback elimination (Section 4), the axon tract over one signaling epoch is represented as a vector channel

where \( \mathbf{X} = (X_1,\dots,X_m)^T \) and \( \mathbf{Y} = (Y_1,\dots,Y_n)^T \) are random vectors whose components correspond to axons or spatial modes, and \( \mathbf{Z} \sim \mathcal{N}(0,\Sigma_{\text{eff}}) \) is multivariate Gaussian noise.

Unlike scalar mutual information, this formulation naturally invites multivariate mutual information (MMI), since dependence is distributed across multiple random variables within each vector.

Multivariate Mutual Information (MMI)

Given a collection of random variables \( (U_1,\dots,U_k) \), the multivariate mutual information (also known as total correlation) is defined as

MMI quantifies the total statistical dependence among the components and vanishes if and only if the variables are independent.

For the axon tract, two natural MMI quantities arise:

• internal dependence among output components ${Y_i}$,

• dependence induced between input and output component collections.

Output MMI of the Tract

Assume $\mathbf{X} \sim \mathcal{N}(0,K_X)$. Then

The multivariate mutual information among the output components $\{Y_1,\dots,Y_n\}$ is

For Gaussian variables,

yielding

This quantity measures the degree to which ephaptic coupling and shared noise induce collective dependence across axons.

Input–Output Multivariate Mutual Information

To characterize tract-level information transfer, consider the MMI between the input vector $\mathbf{X}$ and output vector $\mathbf{Y}$, defined as

For jointly Gaussian $(\mathbf{X},\mathbf{Y})$, with covariance

we obtain

Substitution yields the closed-form expression

Relation to Classical Mutual Information

Classical mutual information satisfies

The MMI decomposes this quantity into contributions from marginal entropies and joint dependence. In particular,

making explicit how tract capacity arises from both signal transmission and collective dependence across axons.

Interpretation

The multivariate mutual information provides a tract-level measure of synergistic coding: ephaptic coupling increases not only channel capacity but also the irreducible dependence among axonal signals. Hence, MMI captures structural information integration beyond what is visible from scalar or pairwise mutual information alone.

Relation to Chung–Chan Network MMI Partitions

Chung–Chan Network MMI

Chung and Chan define the multivariate mutual information of a network $\mathcal{V} = \{V_1,\dots,V_N\}$ via its minimum information partition (MIP). Given a partition $\mathcal{P} = \{C_1,\dots,C_k\}$ of $\mathcal{V}$, the partition information is

and the network MMI is defined as

where the minimum is taken over all nontrivial partitions.

This construction identifies the weakest informational cut of the network and yields a principled measure of global integration.

Axon Tract as a Communication Network

In the present work, the axon tract output \( \mathbf{Y} = (Y_1,\dots,Y_n) \) naturally defines a network whose nodes are individual axons or spatial modes. Ephaptic coupling induces statistical dependence among these nodes, making the Chung–Chan framework directly applicable.

Let $\mathcal{V} = \{Y_1,\dots,Y_n\}$. For any partition $\mathcal{P} = \{C_1,\dots,C_k\}$ of the tract, the partition information is

where $\mathbf{Y}_{C_i}$ denotes the subvector indexed by cluster $C_i$.

Gaussian Evaluation

Since $\mathbf{Y}$ is jointly Gaussian with covariance $K_Y$, each term admits a closed form:

where $K_Y[C_i]$ denotes the principal submatrix indexed by $C_i$.

Thus,

The Chung–Chan network MMI of the tract output is therefore

Relation to Output Total Correlation

The output MMI defined in Section X, \[ I_{{MMI}}(\mathbf{Y}) = {1}{2} ( {_{i=1}^n [K_Y]_{ii}}{ K_Y} ), \] corresponds to the special case of the Chung–Chan framework where the partition $\mathcal{P}$ consists solely of singletons, \( \mathcal{P} = {{Y_1},\dots,{Y_n}}. \)

Hence,

with equality if and only if the singleton partition is the minimum information partition. This occurs when dependence is distributed homogeneously across the tract.

Input–Output Network Partitioning

The Chung–Chan construction also applies to the joint network $\mathcal{V} = \mathbf{X} \cup \mathbf{Y}$. In this case, partitions that separate input and output nodes quantify bottlenecks in tract information transfer.

In particular, the bipartition \( \mathcal{P} = {\mathbf{X},\mathbf{Y}} \) yields

showing that classical mutual information is a special case of network MMI under bipartitioning.

Interpretation for Axon Tracts

Within this framework:

• The minimum information partition identifies the weakest ephaptically coupled cut of the tract.

• A large $I_{{CC}}(\mathbf{Y})$ implies strong global integration across axons, not reducible to independent sub-bundles.

• Classical capacity measures how much information is transmitted, while Chung–Chan MMI measures how irreducibly distributed that information is across the tract.

Thus, the Chung–Chan network MMI provides a natural measure of tract information integration complementary to channel capacity.

Information-Spectrum Reduction to Point-to-Point Tract Channels

From General Information-Spectrum to Point-to-Point Capacity

Han’s information-spectrum framework provides a capacity characterization for channels without assumptions of stationarity, ergodicity, or memorylessness. While originally developed for multi-user channels, the formalism reduces naturally to the single-user (point-to-point) case, which is the relevant abstraction for an axon tract viewed as a distributed channel between an input vector $\mathbf{X}^n$ and an output vector $\mathbf{Y}^n$.

Consider a general point-to-point channel specified by a sequence of conditional distributions \[ W^n(\mathbf{y}^n \mid \mathbf{x}^n), \] with arbitrary (possibly continuous) alphabets. Let ${\mathbf{X}^n}_{n1}$ be an arbitrary input process and $\mathbf{Y}^n$ the corresponding output process induced by the channel.

The normalized information density is defined as

Following Verdu and Han, the achievable rate for a given input process is determined by the liminf in probability

The channel capacity is then

where the supremum is taken over all admissible input processes, including nonstationary and nonergodic ones.

This formulation remains valid even when the channel exhibits spatial coupling, long memory, or blockwise variability—features intrinsic to axon tract propagation.

Reduction to Gaussian Vector Channels

In the present work, the axon tract is modeled as a linear stochastic channel

where:

• $\mathbf{X}^n \in \mathbb{R}^{mn}$ is the stacked input vector across $m$ axons and $n$ temporal samples,

• $H^n$ encodes ephaptic coupling and spatial mixing,

• $\mathbf{Z}^n$ is additive noise, possibly correlated across axons and time.

No assumption of stationarity across $n$ is required. The noise process $\mathbf{Z}^n$ may be nonergodic or mixed, directly aligning with the information-spectrum setting.

For Gaussian inputs and noise, the normalized information density converges in probability, and the information-spectrum capacity reduces to a spectral determinant form:

provided the limit exists.

Thus, the general information-spectrum capacity coincides with the Gaussian mutual information rate even when classical ergodicity assumptions fail.

Application to the Axon Tract Model

In the tract model developed in this paper, the channel input is a vector $\mathbf{X}^n$ of axonal drive signals, and the output $\mathbf{Y}^n$ represents measured or effective field variables along the tract. The channel law is spatially extended and anisotropic, reflecting axonal geometry and ephaptic coupling.

Let $K_{X^n}$ denote the input covariance subject to a global metabolic or energetic constraint

The induced output covariance is

Using the information-spectrum formulation, the achievable rate for a given covariance sequence is

Maximizing over admissible $K_{X^n}$ yields the tract capacity

Importantly, this expression remains valid even if the tract exhibits slowly varying coupling, spatial heterogeneity, or mixed noise regimes, which would invalidate classical Shannon derivations.

Connection to Vector-to-Vector Information Flow

Unlike scalar point-to-point channels, the tract channel maps a vector input to a vector output. Han’s framework accommodates this naturally, since $\mathbf{X}^n$ and $\mathbf{Y}^n$ are treated as arbitrary random objects.

The resulting capacity quantifies aggregate information flow through the tract rather than per-axon throughput. This aligns with the biophysical interpretation that signaling capacity emerges collectively from ephaptically coupled axons, not from isolated fibers.

Moreover, the information-spectrum formulation makes explicit that capacity is determined by the worst-case asymptotic information density, rather than by average mutual information alone. This is particularly relevant for biological systems, where nonstationary fluctuations and rare events may dominate functional performance.

Relation to MMI and Network Integration

The point-to-point capacity derived above complements the multivariate mutual information (MMI) analysis developed earlier. Capacity measures how much information can be reliably conveyed from $\mathbf{X}$ to $\mathbf{Y}$, while MMI quantifies how irreducibly that information is distributed across axonal degrees of freedom.

Within the information-spectrum view, MMI emerges as a structural constraint on achievable capacity: strong output integration (high MMI) implies that capacity cannot be decomposed into independent subchannels. Conversely, weak MMI corresponds to near-block-diagonal covariance structure and additive capacities.

Thus, the tract capacity computed here should be interpreted jointly with the Chung–Chan network MMI as characterizing both quantity and integration of information flow.

Interpretational Summary

The reduction of Han’s general framework to the tract setting yields three key insights:

1. The axon tract is naturally a general point-to-point channel in the information-spectrum sense.

2. Capacity remains well-defined without assuming stationarity, ergodicity, or independence.

3. The resulting expression interfaces cleanly with Gaussian tract models and with multivariate information measures.

This places the present work within a rigorous, modern extension of Shannon theory suitable for structured biological communication systems.

Comparison with Prior Ephaptic Mutual Information Analysis

The present work builds directly on, but also substantially extends, the framework developed in [3], where ephaptic coupling in small axon groups was analyzed primarily through the lens of input–output mutual information computed from simulated Hodgkin–Huxley dynamics. While both works share a common physical substrate and information-theoretic motivation, they differ in modeling abstraction, mathematical scope, and the nature of the information measures employed.

Level of Abstraction and Channel Modeling

In [3], the axon or axon tract is treated as a waveform-to-waveform channel. The transmembrane voltage at an input node and an output node is expanded in a complete orthonormal basis (typically Fourier), and the resulting expansion coefficients are treated as discrete random variables. Mutual information is then computed empirically using histograms derived from repeated numerical simulations.

The present work departs from this explicitly waveform-centric approach by introducing a higher-level vector channel abstraction. Instead of focusing on individual expansion coefficients or their empirical distributions, the tract is modeled as a linear (or linearized) stochastic mapping between high-dimensional input and output vectors, each representing collective axon activity over space and time. This abstraction enables the use of modern multivariate information measures, including multivariate mutual information (MMI) and information-spectrum capacity, which are not directly accessible in the earlier formulation.

Mutual Information versus Capacity

A central distinction lies in the role played by optimization. The prior work explicitly acknowledges that it does not solve the maximization problem required to compute channel capacity, focusing instead on mutual information as a descriptive metric. Capacity is discussed conceptually but not operationally realized.

In contrast, the present work treats capacity as a primary object of study. By invoking Verdu–Han information-spectrum methods and Gaussian vector channel theory, it provides explicit capacity expressions (or lower bounds) under broad conditions, including nonstationarity and spatial coupling. Mutual information appears as an intermediate or structural quantity, whereas capacity emerges as an asymptotic performance limit.

This shift is significant: whereas the earlier work demonstrates how ephaptic coupling can reduce mutual information for fixed signaling distributions, the present work clarifies how ephaptic structure constrains the maximum achievable information flow under optimal signaling.

Role of Ephaptic Coupling

Both works identify ephaptic coupling as a distortion mechanism that tends to reduce pointwise input–output mutual information. In [3], this effect is observed numerically: increased coupling strength leads to reduced mutual information, interpreted as increased synchronization and reduced variability among signals.

The present work reframes this phenomenon in structural information-theoretic terms. Ephaptic coupling is shown to induce statistical dependence across axon outputs, which increases multivariate integration as quantified by MMI. From this perspective, the reduction in pairwise mutual information is not merely a loss, but a redistribution of information from separable channels into integrated collective modes. Thus, ephaptic coupling trades independent capacity for integrated capacity, a distinction not explicit in the earlier analysis.

Dimensionality and Network Perspective

The earlier work primarily treats the $N=2$ axon case, with extensions to higher $N$ discussed formally but not deeply analyzed. Mutual information is computed between aggregate input and output sets, but the internal partitioning structure of the tract is not examined.

The present work explicitly adopts a network information perspective. By using Chung–Chan MMI partitions, it decomposes information flow according to how it is distributed across axonal subsets. This allows one to distinguish between reducible information (attributable to sub-tracts) and irreducible information (emergent at the tract level). In this sense, the present work adds a layer of explanatory power by characterizing not only how much information flows, but how it is integrated.

Mathematical Generality

Methodologically, [3] relies on discrete approximations, empirical probability mass functions, and simulation-based estimation. This approach is well suited for detailed biophysical modeling but becomes cumbersome for large-scale theoretical analysis.

The present work abstracts away many microscopic details to achieve greater generality. Information-spectrum methods allow arbitrary noise statistics, nonergodicity, and spatial heterogeneity, while Gaussian vector models yield closed-form or asymptotic expressions. As a result, the theory applies not only to specific Hodgkin–Huxley simulations, but to a broad class of distributed biological channels.

Interpretational Shift

Finally, there is a conceptual shift in interpretation. The earlier paper frames ephaptic coupling primarily as detrimental to communication, in the sense of reducing mutual information between predefined input and output variables. The present work adopts a more nuanced view: ephaptic coupling reduces decomposability but increases integration. Whether this is "beneficial" or "detrimental" depends on the computational or functional objective of the neural system.

In summary, the present work should be viewed not as a replacement for [3], but as a theoretical extension. The earlier work establishes detailed numerical evidence and biophysical grounding, while the present formulation situates ephaptic axon tracts within modern multivariate and information-spectrum information theory, enabling capacity analysis, integration quantification, and principled comparison with communication networks.

Acknowledgments

This work was prepared with the assistance of LLMs.