Thermodynamic Neurons Overview

• Thermodynamic neurons are a family of neuron-like units defined by explicit thermodynamic formulations such as heat flow, entropy, and charge conservation that differ from classical neuron models.
• They leverage diverse physical mechanisms including adaptive transport, quantum thermal machines, and chemical kinetics to enable computation and self-organization via reversible and irreversible processes.
• Research demonstrates that these models reveal insights into energy efficiency, learning bounds, and critical dynamics at both the single neuron and network levels.

Searching arXiv for papers on thermodynamic neurons and closely related formulations. Thermodynamic neurons are neuron-like units whose state transitions, input–output mappings, or learning dynamics are formulated in explicitly thermodynamic terms rather than solely as abstract digital operations. Across the literature, the term covers several non-equivalent constructions: locally interacting transport units that conserve and dissipate charge, autonomous quantum thermal machines that compute through heat flow, chemical reaction networks that spike and learn, noisy perceptrons constrained by entropy production, and population-level statistical-physics descriptions in which neural states are treated through Gibbs measures, effective temperatures, or thermodynamic limits. Taken together, these works suggest that “thermodynamic neuron” is not a single canonical object but a family of attempts to ground neural computation in conservation, fluctuation, dissipation, adaptation, and nonequilibrium organization (Hylton, 2019, Lipka-Bartosik et al., 2023, Fil et al., 2020, Goldt et al., 2016).

1. Terminological scope

The literature uses the expression in several distinct senses.

Interpretation	Representative object	Representative paper
Adaptive transport unit	Node with sign-segregated charge compartments and dissipative edge adaptation	(Hylton, 2019)
Autonomous thermal logic element	Quantum thermal machine mapping bath temperatures to output temperature	(Lipka-Bartosik et al., 2023)
Autonomous classifier component	Heat-flow logic unit used inside a stochastic Tsetlin-machine-like architecture	(Suderman et al., 24 Jun 2026)
Chemical spiking neuron	Micro-reversible reaction network with Hebbian plasticity	(Fil et al., 2020)
Formal or population-level thermodynamic description	Gibbs measures, effective temperature, thermodynamic limit, or macroscopic order parameters	(Cofré et al., 2020, Merolla et al., 2010, Pietras et al., 2022, Kim et al., 2014)

Two distinctions recur throughout this body of work. First, some papers use thermodynamic language literally, by assigning thermal baths, heat currents, entropy production, or physically conserved quantities to the neuron-like unit; others use thermodynamic language formally, by importing the machinery of Gibbs measures, free energy, entropy, or criticality into spike-train statistics and population dynamics. Second, some works focus on a single neuron-like device, whereas others study only whole-network or population-level descriptions.

A common misconception is that the phrase denotes a standard neuronal primitive analogous to a perceptron or a conductance-based membrane model. Several papers explicitly cut against that reading. “Thermodynamic Neural Network” (Hylton, 2019) states that its model is not a neuron model in the standard artificial-neural-network or conductance-based biological sense; “Thermodynamic bounds on energy use in Deep Neural Networks” (Tkachenko, 13 Mar 2025) treats network-level inference bounds rather than a microscopic neuron Hamiltonian; and the thermodynamic-formalism and synchronization papers use “thermodynamic” for macroscopic statistical characterization rather than literal thermal neurons (Cofré et al., 2020, Kim et al., 2014).

2. Adaptive transport units and the thermodynamic neural network

In “Thermodynamic Neural Network” (Hylton, 2019), the closest analogue to a thermodynamic neuron is the node together with its local compartment structure, stochastic state-selection rule, and edge-adaptation mechanism. Each node $j$ has a potential-like state $e_j$, typically $e_j \in [-1,1]$, while each edge $w_{ij}$ is both a transport conductance and a carrier of sign-segregated charge. The elementary edge quantity is

$q_{ij}^{\pm\pm} = e_{ij}^{\pm} w_{ij}^{\pm\pm},$

and incoming charges are accumulated into four node-local compartments,

$q_j^{\pm\pm} = \sum_i q_{ij}^{\pm\pm}.$

After state selection, output charges are grouped as

$$p_j^{\pm\pm} = e_j w_j^{\pm\pm}, \qquad w_j^{\pm\pm}=\sum_i w_{ij}^{\pm\pm}.$$

Positive and negative charges are independently conserved and “never sum to cancel each other,” which sharply distinguishes the model from ordinary signed-sum neural updates.

The node selects its state by conditional Boltzmann sampling,

$$P_j(e_j \mid e\neq e_j,w,q) = \frac{\exp(-\beta H_j(e_j))}{Z_j},$$

with local gating factors

$f_j^\pm(e_j)=h(\mp e_j),$

where $h$ is the Heaviside step function. These factors determine which pair of compartments is active. The simplified node energy has the form

$e_j$0

so the chosen sign of $e_j$1 lowers energy when it supports transport between complementary compartments. The resulting ordering is antiferromagnetic-like: connected nodes tend to favor opposite potentials.

A central contribution of the model is the separation between reversible equilibration and irreversible adaptation. If a node’s energy fluctuations are larger than temperature, it performs reversible updates: it samples $e_j$2, exchanges charges, and explores configurations without changing $e_j$3. If local energy fluctuations are smaller than temperature, the node is treated as locally equilibrated and performs irreversible updates, dissipating residual charges

$e_j$4

into nearby edge weights. The network therefore operates on two timescales: fast reversible equilibration of node potentials and slow irreversible structural adaptation of transport conductances. Externally driven networks evolve to efficiently connect external positive and negative potentials, and the paper’s explicit broader claim is that the transport and dissipation of conserved physical quantities drives the self-organization of open thermodynamic systems. It is therefore more accurate to classify this model as a self-organizing transport network and a broader thermodynamic theory of adaptive open systems than as a canonical neuron model.

3. Autonomous thermal machines and logic-based thermodynamic neurons

A more literal use of the term appears in “Thermodynamic Computing via Autonomous Quantum Thermal Machines” (Lipka-Bartosik et al., 2023), where a thermodynamic neuron is an autonomous quantum thermal machine that computes by setting input bath temperatures, relaxing to a non-equilibrium steady state, and reading the output from the temperature of an auxiliary finite-size reservoir. The machine is built from a collector and a modulator. The collector forms a virtual qubit whose virtual inverse temperature is

$e_j$5

which is the thermal analogue of a perceptron’s weighted linear combination. The modulator then maps this internal variable to a nonlinear output through the steady-state output reservoir temperature. In the small-$e_j$6 limit the output approaches a sigmoid,

$e_j$7

Within this construction, a single thermodynamic neuron can implement any linearly separable Boolean function; the paper discusses NOT, NOR, and 3-MAJORITY explicitly, and then argues that networks of such neurons can perform arbitrary Boolean computation.

“Interpretable rule-based learning in an autonomous thermodynamic network” (Suderman et al., 24 Jun 2026) pushes this architecture toward full logical inference. There, thermodynamic neurons are again autonomous quantum thermal machines, now used as AND, OR, and NOT gates inside a stochastic Tsetlin-machine-like classifier. The collector forms a virtual temperature, the modulator imposes a nonlinear saturating response, and the logical output is read by a first-passage rule on a signal qubit coupled to the finite output bath. The basic steady-state population relation is

$e_j$8

and entropy production is written as

$e_j$9

The paper emphasizes that reliable computation does not arise from exact logical operations, but from thresholding and redundancy. With redundancy $e_j \in [-1,1]$0, the thermodynamic classifier achieves testing accuracies statistically comparable to the standard Tsetlin machine across several tabular benchmarks, and the authors explicitly present this as autonomous, interpretable learning driven by noisy dissipative dynamics rather than by deterministic logical circuitry.

These thermal-machine models narrow the meaning of thermodynamic neuron to a specific physical object: a heat-flow device that computes through autonomous relaxation. They are also the clearest examples in which the analogy to a perceptron is exact at the architectural level: bath temperatures play the role of inputs, qubit-energy design fixes effective weights, virtual temperature is the pre-activation, and the finite reservoir plus modulator supplies the nonlinear activation.

4. Chemical, spiking, and material realizations

In “A thermodynamically consistent chemical spiking neuron capable of autonomous Hebbian learning” (Fil et al., 2020), the neuron is realized as a chemical reaction network. Channel-specific species $e_j \in [-1,1]$1 represent inputs, $e_j \in [-1,1]$2 represent synaptic weights, $e_j \in [-1,1]$3 acts like membrane potential, and the activated species $e_j \in [-1,1]$4 acts as output spike and plasticity signal. The model is built from micro-reversible elementary reactions with mass-action kinetics. Input transduction and weighted conversion to $e_j \in [-1,1]$5 are mediated by $e_j \in [-1,1]$6; coincidence of $e_j \in [-1,1]$7 and $e_j \in [-1,1]$8 creates more $e_j \in [-1,1]$9, implementing Hebbian plasticity; and leak terms remove both $w_{ij}$0 and $w_{ij}$1. The paper’s central computational result is that efficient computation of time-correlations requires a highly nonlinear activation function, implemented by an $w_{ij}$2-step cooperative binding chain. Frequency-bias learning is comparatively easy, whereas temporal-correlation learning requires sharp thresholding and incurs explicit resource costs because increasing $w_{ij}$3 sequesters more of the internal-state resource.

A different route to thermodynamic spiking appears in “The thermodynamic temperature of a rhythmic spiking network” (Merolla et al., 2010). There the single neuron is an integrate-and-fire unit observed over a fixed window, and excitatory and inhibitory Poisson inputs play the role of temperature. For subthreshold baseline input, the spiking “bandgap” is approximated by

$w_{ij}$4

With a global inhibitory rhythm imposing synchronous windows, the noisy spiking neuron behaves like a stochastic binary unit. Matching the spike-probability curve to a logistic produces an effective temperature

$w_{ij}$5

and a network of such neurons can implement a single-layer Boltzmann machine without learning. Here thermodynamic neurons are not physical thermal machines; rather, they are rhythmic spiking neurons endowed with an effective thermodynamic temperature through synaptic noise.

Biophysical energetics enters more directly in “Temperature Effects on Information Capacity and Energy Efficiency of Hodgkin-Huxley Neuron” (Wang et al., 2015). That paper studies a single-compartment Hodgkin–Huxley neuron under noisy synaptic-like input and treats temperature as a control variable that jointly modulates spike generation, information rate, and energetic cost. The central efficiency measure is

$w_{ij}$6

Both information rate and energy efficiency are nonmonotonic in temperature and each has an optimum, but the optima do not coincide. At the energy-efficiency optimum, the neuron still retains $w_{ij}$7–$w_{ij}$8 of its maximal information rate. This line of work does not redefine the neuron as a thermal machine, but it does treat the neuron as a thermodynamically constrained device whose computational merit is inseparable from energy expenditure.

At the hardware-material level, “Collective dynamics and long-range order in thermal neuristor networks” (Zhang et al., 2023) studies VO$w_{ij}$9-based spiking devices coupled through heat diffusion. Individual devices spike through charging, Joule heating, insulator-to-metal transition, discharge, cooling, and reset; neighboring devices interact thermally through the substrate. Arrays exhibit synchronized oscillations, clusters, propagating waves, and long-range order. The paper’s striking conclusion is that the observed long-range order does not arise from criticality but from time non-local thermal response. For reservoir computing, the original interacting reservoir reaches $q_{ij}^{\pm\pm} = e_{ij}^{\pm} w_{ij}^{\pm\pm},$0 on MNIST, the no-interaction version reaches $q_{ij}^{\pm\pm} = e_{ij}^{\pm} w_{ij}^{\pm\pm},$1, and a reduced-memory version reaches $q_{ij}^{\pm\pm} = e_{ij}^{\pm} w_{ij}^{\pm\pm},$2, supporting the claim that efficient performance does not require criticality or long-range order in this platform.

5. Thermodynamic bounds on learning and inference

A separate strand of work treats thermodynamic neurons as noisy learners whose synapses or activation states obey explicit thermodynamic constraints. In “Stochastic Thermodynamics of Learning” (Goldt et al., 2016), the model is a noisy perceptron with stochastic weights and stochastic binary outputs. The central bound for a single learned label is

$q_{ij}^{\pm\pm} = e_{ij}^{\pm} w_{ij}^{\pm\pm},$3

and the general multi-weight, multi-sample version is

$q_{ij}^{\pm\pm} = e_{ij}^{\pm} w_{ij}^{\pm\pm},$4

This motivates a learning efficiency

$q_{ij}^{\pm\pm} = e_{ij}^{\pm} w_{ij}^{\pm\pm},$5

The paper’s conceptual move is to identify learning as a nonequilibrium physical process: synapses are noisy memory variables, outputs are thermal two-state variables, and information acquisition is bounded by thermodynamic cost.

“Thermodynamic efficiency of learning a rule in neural networks” (Goldt et al., 2017) extends this logic to teacher–student learning and generalization. For a single synapse, the sharpened steady-state bound is

$q_{ij}^{\pm\pm} = e_{ij}^{\pm} w_{ij}^{\pm\pm},$6

with efficiency

$q_{ij}^{\pm\pm} = e_{ij}^{\pm} w_{ij}^{\pm\pm},$7

In the thermodynamic limit, generalization is expressed through the order parameters

$q_{ij}^{\pm\pm} = e_{ij}^{\pm} w_{ij}^{\pm\pm},$8

and the generalization error

$q_{ij}^{\pm\pm} = e_{ij}^{\pm} w_{ij}^{\pm\pm},$9

The paper analyzes Hebbian, Perceptron, and AdaTron learning, and reports that maximal efficiency is only about $q_j^{\pm\pm} = \sum_i q_{ij}^{\pm\pm}.$0. It also identifies a breakdown of AdaTron learning above the critical normalized learning rate $q_j^{\pm\pm} = \sum_i q_{ij}^{\pm\pm}.$1. The broader implication is that low dissipation alone does not guarantee good learning: informational efficiency of the rule itself also matters.

At the deep-network level, “Thermodynamic bounds on energy use in Deep Neural Networks” (Tkachenko, 13 Mar 2025) shifts attention from learning dynamics to inference cost. The paper distinguishes digital implementations, dynamic analog systems, and quasi-static analog systems. Its main lower bound for inference is

$q_j^{\pm\pm} = \sum_i q_{ij}^{\pm\pm}.$2

with the corresponding negentropy bound

$q_j^{\pm\pm} = \sum_i q_{ij}^{\pm\pm}.$3

The key message is that linear transformations in a feedforward DNN can in principle be implemented reversibly in suitable analog substrates, whereas the nonlinear activation step introduces unavoidable irreversibility because the binary activation variables $q_j^{\pm\pm} = \sum_i q_{ij}^{\pm\pm}.$4 are “nothing but regular digital bits.” The same source explicitly states, however, that it does not provide a thermodynamic treatment of training, does not introduce an explicit Hamiltonian for the network, and does not derive neuron-level equilibrium statistical mechanics. In this sense, it is relevant to thermodynamic neurons mainly by identifying the likely locus of irreversibility: thresholded activation rather than linear summation.

6. Population thermodynamics, formal analogies, and open controversies

Several influential papers use thermodynamic language at the level of neural populations rather than individual thermodynamic devices. In “Thermodynamics for a network of neurons: Signatures of criticality” (Tkacik et al., 2014), effective energy is defined directly from neural-state probability,

$q_j^{\pm\pm} = \sum_i q_{ij}^{\pm\pm}.$5

with entropy

$q_j^{\pm\pm} = \sum_i q_{ij}^{\pm\pm}.$6

For retinal ganglion-cell populations, the estimated entropy–energy relation approaches

$q_j^{\pm\pm} = \sum_i q_{ij}^{\pm\pm}.$7

that is, $q_j^{\pm\pm} = \sum_i q_{ij}^{\pm\pm}.$8 over a broad range, which the authors interpret as proximity to an unusual kind of critical point. “Thermodynamic Formalism in Neuronal Dynamics and Spike Train Statistics” (Cofré et al., 2020) gives the rigorous symbolic-dynamics version of this program: spike trains are modeled by Gibbs measures, pressure, transfer operators, and the variational principle

$q_j^{\pm\pm} = \sum_i q_{ij}^{\pm\pm}.$9

In this usage, thermodynamic neurons are not literal heat devices; they are spike-generating systems whose statistics can be cast in the formal language of equilibrium states and Gibbs distributions.

A related but distinct formalism appears in “Neurons as an Information-theoretic Engine” (Shimazaki, 2015), where a maximum-entropy population model yields the exact entropy balance

$$p_j^{\pm\pm} = e_j w_j^{\pm\pm}, \qquad w_j^{\pm\pm}=\sum_i w_{ij}^{\pm\pm}.$$0

Internal modulation of gain is then interpreted as an information-theoretic cycle, and the optimal entropic efficiency of the ideal cycle is

$$p_j^{\pm\pm} = e_j w_j^{\pm\pm}, \qquad w_j^{\pm\pm}=\sum_i w_{ij}^{\pm\pm}.$$1

Similarly, “Thermodynamic Order Parameters and Statistical-Mechanical Measures for Characterization of the Burst and Spike Synchronizations of Bursting Neurons” (Kim et al., 2014) uses “thermodynamic” for macroscopic order parameters based on the instantaneous population firing rate $$p_j^{\pm\pm} = e_j w_j^{\pm\pm}, \qquad w_j^{\pm\pm}=\sum_i w_{ij}^{\pm\pm}.$$2, not for literal thermal neurons. These papers are best read as structured thermodynamic analogies grounded in maximum-entropy geometry and nonequilibrium statistical mechanics, rather than as proposals for microscopic thermodynamic neuron hardware.

Another sense of thermodynamic neural description is the thermodynamic limit of large spiking populations. “Exact finite-dimensional description for networks of globally coupled spiking neurons” (Pietras et al., 2022) proves that, in the limit $$p_j^{\pm\pm} = e_j w_j^{\pm\pm}, \qquad w_j^{\pm\pm}=\sum_i w_{ij}^{\pm\pm}.$$3, globally coupled theta/QIF populations with Cauchy white noise or Cauchy-Lorentz heterogeneity admit an exact six-dimensional macroscopic description, asymptotically converge to the two-dimensional Ott–Antonsen/Lorentzian manifold for $$p_j^{\pm\pm} = e_j w_j^{\pm\pm}, \qquad w_j^{\pm\pm}=\sum_i w_{ij}^{\pm\pm}.$$4, and reduce to a three-dimensional Watanabe–Strogatz form in the identical noise-free case. This is a thermodynamic-limit theory in the statistical-mechanical sense: the macrostate of the neural population closes exactly, but it is not an equilibrium thermodynamics of single neurons.

The literature also contains genuinely biophysical thermodynamic analyses of neurons. “The Heat of Nervous Conduction: A Thermodynamic Framework” (Lichtervelde et al., 2019) revisits the classical condenser theory of action-potential heat and derives the membrane free-energy contribution

$$p_j^{\pm\pm} = e_j w_j^{\pm\pm}, \qquad w_j^{\pm\pm}=\sum_i w_{ij}^{\pm\pm}.$$5

With an inner surface charge density $$p_j^{\pm\pm} = e_j w_j^{\pm\pm}, \qquad w_j^{\pm\pm}=\sum_i w_{ij}^{\pm\pm}.$$6 more negative than the outer surface, the predicted heat release enters the experimental range. This is one of the clearest examples in which literal thermodynamic variables—free energy, entropy, reversible heat release and absorption—are attached to real neuronal membranes.

Finally, some proposals are more revisionist. “Neuronal electricality founded in murburn-thermodynamic principles: 1. Background and basic theoretical formulation” (Manoj et al., 15 Apr 2026) introduces an Electron Holding Potential $$p_j^{\pm\pm} = e_j w_j^{\pm\pm}, \qquad w_j^{\pm\pm}=\sum_i w_{ij}^{\pm\pm}.$$7 and derives a reaction–transport–relaxation equation

$$p_j^{\pm\pm} = e_j w_j^{\pm\pm}, \qquad w_j^{\pm\pm}=\sum_i w_{ij}^{\pm\pm}.$$8

The paper presents this as a chemically grounded alternative to ion-centric electrophysiology and explicitly defers validation and falsification to a second part. This suggests that frontier uses of “thermodynamic neuron” still include proposals that challenge standard electrophysiological foundations, but such proposals remain theories in need of further testing rather than established consensus.

Taken together, the field supports a restrained conclusion. Thermodynamic neurons are best understood as a family of research programs that attempt to re-express neural computation in terms of heat flow, conserved transport, stochastic thermodynamics, chemical kinetics, or statistical-mechanical state structure. Some of these programs produce concrete physical devices; others yield formal descriptions of populations or spike trains; and several explicitly deny that they are standard neuron models. The term therefore names a domain of inquiry rather than a settled architecture.