Springs and a stopwatch: neural units with time-dependent multifunctionality
Stephen Whitelam



=1
















#1#1
#1Eq. (<ref>)
#1Eqns. (<ref>)
#1(<ref>)

#1Fig. <ref>
#1Figure <ref>
#1Figs. <ref>

#1Section <ref>

#̧1 <cit.>
#1 <cit.>
#1Ref. <cit.>
#1Refs. <cit.>

#1⟨#1 ⟩


    ck_ BT_0t_ fswhitelam@lbl.govMolecular Foundry, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
        Several branches of computing use a system's physical dynamics to do computation. We show that the dynamics of an underdamped harmonic oscillator can perform multifunctional computation, solving distinct problems at distinct times within a single dynamical trajectory. Oscillator computing usually focuses on the oscillator's phase as the information-carrying component. Here we focus on the time-resolved amplitude of an oscillator whose inputs influence its frequency, which has a natural parallel as the activity of a time-dependent neural unit. Because the activity of the unit at fixed time is a nonmonotonic function of the input, the unit can solve nonlinearly-separable problems such as XOR. Because the activity of the unit at fixed input is a nonmonotonic function of time, the unit is multifunctional in a temporal sense, able to carry out distinct nonlinear computations at distinct times within the same dynamical trajectory. Time-resolved computing of this nature can be done in or out of equilibrium, with the natural time evolution of the system giving us multiple computations for the price of one.
        Springs and a stopwatch: neural units with time-dependent multifunctionality
            Stephen Whitelam
            
        ============================================================================
        
        
        
        § INTRODUCTION
         Computing is done by physical processesb̧ȩņņȩţţ1̧9̧8̧2̧ţḩȩŗm̧o̧ḑy̧ņa̧m̧i̧çş,̧ļa̧ņḑa̧u̧ȩŗ1̧9̧9̧1̧i̧ņf̧o̧ŗm̧a̧ţi̧o̧ņ,̧w̧o̧ļp̧ȩŗţ2̧0̧1̧9̧şţo̧çḩa̧şţi̧ç. Classical computing uses the movement of electrons in silicon chips to perform logical operationsçȩŗu̧z̧z̧i̧2̧0̧0̧3̧ḩi̧şţo̧ŗy̧; quantum computing uses superposition and entanglement in qubits to process informationḩo̧ŗo̧w̧i̧ţz̧2̧0̧1̧9̧q̧u̧a̧ņţu̧m̧; neuromorphic computing mimics the neural and synaptic activities of the human brainşçḩu̧m̧a̧ņ2̧0̧1̧7̧şu̧ŗv̧ȩy̧; echo-state networks use dynamic reservoirs of neural activity to process sequencesşu̧ņ2̧0̧2̧0̧ŗȩv̧i̧ȩw̧; analog computing uses the continuous variation of electrical or mechanical signals to solve problemsm̧a̧çļȩņņa̧ņ2̧0̧0̧7̧ŗȩv̧i̧ȩw̧,̧u̧ļm̧a̧ņņ2̧0̧1̧3̧a̧ņa̧ļo̧ģ,̧çşa̧b̧a̧2̧0̧2̧0̧ço̧u̧p̧ļȩḑ; and thermodynamic computing uses the tendency of physical systems to evolve toward thermal equilibrium to do calculationsço̧ņţȩ2̧0̧1̧9̧ţḩȩŗm̧o̧ḑy̧ņa̧m̧i̧ç,̧ḩy̧ļţo̧ņ2̧0̧2̧0̧ţḩȩŗm̧o̧ḑy̧ņa̧m̧i̧ç,̧a̧i̧f̧ȩŗ2̧0̧2̧3̧ţḩȩŗm̧o̧ḑy̧ņa̧m̧i̧ç. 
        
        Here we show that the explicit time dependence of a physical dynamics permits multifunctional computation, with a single device able to perform multiple distinct calculations in the course of a single trajectory. To illustrate this idea we consider the time-dependent dynamics of a continuous-valued underdamped oscillator. Oscillators can be realized experimentally in many waysv̧o̧ņ1̧9̧5̧7̧ņo̧ņ,̧çşa̧b̧a̧2̧0̧2̧0̧ço̧u̧p̧ļȩḑ,̧çi̧ļi̧b̧ȩŗţo̧2̧0̧1̧7̧ȩx̧p̧ȩŗi̧m̧ȩņţş, including by mechanical cantileversḑa̧ģo̧2̧0̧2̧1̧i̧ņf̧o̧ŗm̧a̧ţi̧o̧ņ,̧ḑa̧ģo̧2̧0̧2̧2̧v̧i̧ŗţu̧a̧ļ and electrical circuitsw̧a̧ņģ2̧0̧1̧9̧o̧i̧m̧,̧çḩo̧u̧2̧0̧1̧9̧a̧ņa̧ļo̧ģ,̧m̧ȩļa̧ņşo̧ņ2̧0̧2̧3̧ţḩȩŗm̧o̧ḑy̧ņa̧m̧i̧ç. Computing with oscillators is a concept that dates back to the 1950sv̧o̧ņ1̧9̧5̧7̧ņo̧ņ,̧ģo̧ţo̧1̧9̧5̧9̧p̧a̧ŗa̧m̧ȩţŗo̧ņ. Most examples of oscillator-based computing focus on the phase of the oscillator as a means of carrying information, and aim to find oscillatory ground states in networks of interacting oscillatorsçşa̧b̧a̧2̧0̧2̧0̧ço̧u̧p̧ļȩḑ,̧b̧o̧ņņi̧ņ2̧0̧2̧2̧ço̧u̧p̧ļȩḑ. 
        
        Here we consider the time-resolved amplitude of an oscillator whose inputs influence its frequency, which has a natural parallel as the activity of a time-dependent neural unit. The motivation for this choice is twofold. First, because the activity of the unit at fixed time is a nonmonotonic function of the input, a single unit can solve nonlinearly-separable problems such as XOR, making it more expressive than standard artificial neurons. Second, because the activity of the unit at fixed input is a nonmonotonic function of time, the unit is multifunctional in a temporal sense, able to carry out different nonlinear computations at different times within the same dynamical trajectory. 
        
        Units that are a nonmonotonic function of their input are more expressive than standard artificial neurons. For example, logic operations such as XOR cannot rendered linearly separable by a standard perceptron unitŗu̧m̧ȩļḩa̧ŗţ1̧9̧8̧6̧ļȩa̧ŗņi̧ņģ, but can be solved by oscillator unitsģi̧ḑo̧ņ2̧0̧2̧0̧ḑȩņḑŗi̧ţi̧ç,̧ņo̧ȩļ2̧0̧2̧1̧b̧i̧o̧ļo̧ģi̧ça̧ļļy̧. Consider two binary variables, I_1,2=0,1: the oscillatory function f(I_1,I_2)=sin(π(I_1+I_2)/2) makes the XOR problem linearly separable, because f(0,0)=f(1,1)=0 and f(0,1)=f(1,0)=1. Some neurons in the brain operate in an oscillatory wayģi̧ḑo̧ņ2̧0̧2̧0̧ḑȩņḑŗi̧ţi̧ç, an observation that has motivated other authors to show that neural networks built from oscillator units are highly expressiveȩf̧f̧ȩņb̧ȩŗģȩŗ2̧0̧2̧2̧f̧u̧ņçţi̧o̧ņa̧ļ.
        
        Moreover, temporal oscillations permit a single dynamical neural unit to be multifunctional, performing multiple computations in the course of a single dynamical trajectory. We show that a single oscillator neuron can act as  all of the elementary logic gates, depending on the time at which we measure its output, and can be trained by gradient descent to do distinct classification tasks at distinct times. A device built from such units could perform multiple computations in a single dynamical trajectory, requiring only a single set of parameters to do multiple tasks. 
        
        
        
        § TIME-RESOLVED COMPUTATION IS MULTIFUNCTIONAL
         In more detail, consider a continuous degree of freedom S(t) of unit mass that evolves according to the underdamped Langevin dynamics S̈=-γṠ -∂_S U+η(t). Here γ is a friction coefficient, U is a potential, and η is a noise term. We will consider the harmonic potential U= 1/2 (ω_0^2+I) S^2, where ω_0^2 is the fundamental frequency of the oscillator and I is the input signal (the input signal could in general be time-dependent, but here we consider a constant input). The input therefore influences the spring constant of the harmonic potential. In general, thermal noise provides an important mechanism for driving dynamical evolution and allowing probabilistic computationa̧i̧f̧ȩŗ2̧0̧2̧3̧ţḩȩŗm̧o̧ḑy̧ņa̧m̧i̧ç. Here, for simplicity, we assume the low-noise limit, and so the oscillator evolves according to the equation
        S̈+γṠ +(ω_0^2+I) S=0.
        
        We assume that the oscillator is prepared with initial conditions S(0)=1 and Ṡ(0)=0. Then S satisfies
        
        S(t) = e^-γ t/2( cosΩ t+ γ/2 ΩsinΩ t),
        
        where 
        Ω^2 ≡ω_0^2-γ^2/4+I
        
        is a function of the oscillator's intrinsic parameters and the input I. The input therefore influences the period 2 π/Ω of the oscillator (we assume the underdamped regime, where Ω^2>0). At fixed time the output of the unit is a nonmonotonic function of I, and at fixed I the output of the unit is a nonmonotonic function of t. These two properties give this neural unit the ability to solve nonlinearly separable problems, and to do so in a temporally multifunctional way. 
        
        A neural unit that evolves in time can perform computations in a multifunctional way. Consider a binary input I=0,1. In fig1(a) we show the neuron output sol as a function of time t for the two possible values of I. With a horizontal decision boundary placed at zero, the neuron can perform any of the elementary one-bit operations, depending on observation time: identity (I → I) at t_1; erase to one (I → 1) at t_2; NOT or invert (I →I̅) at t_3; and erase to zero (I → 0) at t_4. The same operations are also performed at other times in the unit's trajectory. 
        
        
        Conventional physical models of processes such as bit erasure consider the manipulation of a degree of freedom in an external potential, with work expendedļa̧ņḑa̧u̧ȩŗ1̧9̧6̧1̧i̧ŗŗȩv̧ȩŗşi̧b̧i̧ļi̧ţy̧,̧şa̧ģa̧w̧a̧2̧0̧1̧2̧ţḩȩŗm̧o̧ḑy̧ņa̧m̧i̧çş,̧z̧u̧ļķo̧w̧şķi̧2̧0̧1̧4̧o̧p̧ţi̧m̧a̧ļ,̧p̧ŗo̧ȩşm̧a̧ņş2̧0̧2̧0̧f̧i̧ņi̧ţȩ,̧ḑa̧ģo̧2̧0̧2̧2̧ḑy̧ņa̧m̧i̧çş. Here, energy must be input in order to set the initial state of the unit and impose and maintain the input to the neuron (which sets the potential spring constant). But after setting the initial condition no work is performed: computations are done by the system's natural dynamics [Single operations and multiple operations incur the same energy cost, with the caveat that the input I to the neuron must be maintained long enough to observe these operations.]. 
        
        Now consider two binary input degrees of freedom I_1,2=0,1, and construct the input I=I_1+I_2. I can take three values, 0,1, or 2. In fig1(b) we show the neuron output sol as a function of time t for the three possible values of I. With the horizontal decision boundary placed as shown, the neuron can function as any of the elementary logic gates, depending on observation time. If we observe the unit at time t_1 it functions as an AND gate; at time t_2, an OR gate; and at time t_3, an XOR gate. At different times the same unit can also function as the inverted versions of these gates, NAND, XNOR, and NOR, the latter shown at time t_4.
        
        The equilibrium state of the unit is S(t →∞)=0, regardless of the value of I, and so in equilibrium the unit has no computational ability. In general, we can do time-resolved computation in or out of equilibrium, but doing it out of equilibrium means that we do not need to wait for equilibrium to be attained.
        
        
        
        § TRAINING OSCILLATOR NEURONS TO PERFORM MULTIFUNCTIONAL COMPUTATION
         Dynamical oscillator neurons can be trained by gradient descent to achieve distinct tasks at distinct times. To illustrate this point we consider the MNIST data setļȩçu̧ņ1̧9̧9̧8̧ģŗa̧ḑi̧ȩņţ, which consists of greyscale images of handwritten digits on a grid of N=28 × 28 pixels, each digit belonging to one of ten classes C ∈ [0,9]; see fig2(a). We wish to construct a classifier that discriminates 0s (class C=0) from 1s (class C=1) at observation time t_1; discriminates 4s from 5s at time t_2; and discriminates 8s from 9s at time t_3. We choose observation times t_1, t_2, and t_3 such that Ω_0 t_k=- arctan(2 Ω_0/γ)+ (k+3) π, where Ω_0^2 ≡ω_0^2-γ^2/4, corresponding to three of the roots of the function sol with I=0. 
        
        A loss function suitable for this multi-time classification task is
        ϕ= ∑_k=1,2,3∑_α∈ C_4k,C_4k+1 (s_α(t_k)-s̅_α)^2.
        
        Here α labels MNIST digits, and the inner sum runs over training-set digits in classes C_4k and C_4k+1. We have defined s_α(t_k) ≡ e^γ t/2S_α(t_k); the exponential scaling cancels the exponential decay of the solution sol, ensuring that each term in phi is of equal importance. S_α(t_k) denotes the value of sol at time t_k when shown MNIST digit α. We consider N connections between digit and neuron: when shown digit α, the neuron input is I_α = ∑_i=1^N θ_i I_i,α, where I_i,α is the value of the i^ th pixel of MNIST digit α and the N quantities θ_i are adjustable weights. In what follows we use the notation Ω_α^2 ≡ω_0^2-γ^2/4+I_α. The target activity in phi is s̅_α=∓ 1/2, where the negative or positive sign is chosen if digit α is a member of an even-numbered class (0,4,8) or an odd-numbered class (1,5,9), respectively.
        
        To train the neuron to solve this multi-time classification task we adjust its input weights by gradient descent on ϕ. Weights are initially chosen randomly, θ_i ∼ N(0,10^-4); we then iterate the equation θ_i →θ_i - α_0 ∂ϕ/∂θ_i for all N weights θ_i. Here α_0 is the learning rate and 
        ∂ϕ/∂θ_i= ∑_k=1,2,3∑_α∈ C_4k,C_4k+1 (s_α(t_k)-s̅_α) g(t_k,Ω_α) I_i,α,
        
        where
        
        g(t,Ω) ≡γ t/2 Ω^2cosΩ t -1/Ω( t+ γ/2 Ω^2) sinΩ t.
        
        We take the inner sum in grad over stochastically-chosen minibatches of N_ batch=300 digits, split equally between classes C_4k and C_4k+1. We set the learning rate to α_0 = 10^-3 (3 N_ batch)^-1 [It could happen during training that Ω_α^2 becomes negative for certain α, indicating overdamping, in which case we must make the replacement Ω→ i Ω in sol and gee. This did not happen in the simulations reported here.]. 
        
        Results are shown in fig2(b). We plot time traces of neuron activity when shown 10 test-set digits (unseen during training) in each class. Classes are indicated by color. The neuron responds differently to each class, and can (imperfectly) distinguish the required classes at the required times: on the right of the panel we show histograms of neuron activity, using all digits of that class in the MNIST test set, at the designated observation times.
        
        As with standard neural units, collections of oscillator neurons are more expressive than individual neurons. Multiple oscillator neurons can have a total activity that is not periodic in time, even if they do not interact. In fig2(c) we show the analog of panel (b) for a collection of 4 noninteracting neurons. Each has the same parameters ω_0 and γ as the single neuron of panel (b), and each has N input weights. We impose the loss function ϕ, with S_α(t) replaced by  1/4∑_i=1^4 S^(i)_α(t). Here S^(i)_α(t) is the output of neuron i at time t when shown MNIST digit α. As before, we adjust the input weights by gradient descent on the loss. 
        
        The time traces and histograms of fig2(c) show the trained neural gas to be more expressive than the trained individual neuron, better distinguishing the required classes at the designated observation time (the loss values for the individual neuron and the gas are shown in fig2(d)). In panel (e) we show histograms of neural activity at a time t_3 for which the neural gas is trained to discriminate 8s and 9s but not 0s and 1s; there, it cannot distinguish 0s and 1s.
        
        
        
        § CONCLUSIONS
         Several branches of computing use the natural evolution of a physical system to do calculations. We have shown that the dynamics of an underdamped harmonic oscillator can perform multifunctional computation, with the same physical system able to solve distinct problems at distinct times within a single dynamical trajectory. The idea proposed here, a form of analog computing, is a nonstandard form of oscillator computing: the latter usually focuses on the information contained within the phase of an oscillator, and seeks to identify the ground-state phases of coupled oscillators. Here we have considered the time-resolved amplitude of an oscillator whose inputs influence its frequency, which we interpret as a time-dependent neural unit. The activity of the unit at fixed time is a nonmonotonic function of the input, and so it can solve nonlinearly-separable problems such as XOR. The activity of the unit at fixed input is a nonmonotonic function of time, and so the unit is multifunctional in a temporal sense, able to carry out distinct nonlinear computations at distinct times within the same dynamical trajectory. We have shown that such units can function as all of the elementary logic gates, depending on observation time, and have shown how to train such units by gradient descent to perform distinct classification tasks at distinct times.
        
        A single temporal oscillator neuron can do nonlinear computation in a multifunctional way, and is a natural building block for neural networks and other devices. Neural networks can be built from interacting oscillators just as they are built from standard artificial neuronsȩf̧f̧ȩņb̧ȩŗģȩŗ2̧0̧2̧2̧f̧u̧ņçţi̧o̧ņa̧ļ. Oscillator neural nets designed for multifunctional computation would have considerable computational power, particularly if the connections between neurons were designed to make efficient use of the multiple distinct computations done by neurons at distinct times.  
        
        The idea described here does not require thermal noise but could be applied in its presence: in fields such as thermodynamic computing, thermal noise provides an important mechanism for driving dynamical evolution and allowing probabilistic computationa̧i̧f̧ȩŗ2̧0̧2̧3̧ţḩȩŗm̧o̧ḑy̧ņa̧m̧i̧ç. The results of this paper suggest extending the idea of thermodynamic computing to consider the time-resolved dynamics of devices that already exist, such as networks of interacting oscillators printed on circuit boardsw̧a̧ņģ2̧0̧1̧9̧o̧i̧m̧,̧çḩo̧u̧2̧0̧1̧9̧a̧ņa̧ļo̧ģ,̧m̧ȩļa̧ņşo̧ņ2̧0̧2̧3̧ţḩȩŗm̧o̧ḑy̧ņa̧m̧i̧ç. 
        
        Time-resolved multifunctional computing provides a way of carrying out multiple nonlinear computations within a single dynamical trajectory of a device. This idea could help reduce the number of parameters or the size of devices needed to do computation, with the natural time evolution of a device giving us, in effect, multiple computations for the price of one.
        
        
        
        § ACKNOWLEDGMENTS.
         I thank Isaac Tamblyn for comments on the paper. This work was done at the Molecular Foundry at Lawrence Berkeley National Laboratory, supported by the Office of Basic Energy Sciences of the U.S. Department of Energy under Contract No. DE-AC02–05CH11231.
        
        
        
        
        
        
        
        
        
        34
        fxundefined [1]
         ifx#1
        fnum [1]
          #1firstoftwo
         secondoftwo
         
        fx [1]
          #1firstoftwo
         secondoftwo
         
        
        
        
        
        
        noop [0]secondoftwo
        
        ref[1]@startlink#1@href
        href[1]#1@endlink
        anitize@url [0] `
        12 `$12
          `&12 `#12 `1̂2 `_12 `%12
        startlink[1]
        endlink[0]
        
        rl [1]href #1
        
        
        
        
        
        
        
        
        
        
        
        
        @bib@innerbibempty
        
         [Bennett(1982)]bennett1982thermodynamics
          authorauthorC. H. Bennett, @noop journaljournalInternational Journal of Theoretical Physics volume21, pages905 (year1982)NoStop
         [Landauer(1991)]landauer1991information
          authorauthorR. Landauer, @noop journaljournalPhysics Today volume44, pages23
          (year1991)NoStop
         [Wolpert(2019)]wolpert2019stochastic
          authorauthorD. H. Wolpert, @noop journaljournalJournal of Physics A: Mathematical and Theoretical volume52, pages193001 (year2019)NoStop
         [Ceruzzi(2003)]ceruzzi2003history
          authorauthorP. E. Ceruzzi, @noop titleA history of modern
          computing (publisherMIT press, year2003)NoStop
         [Horowitz and Grumbling(2019)]horowitz2019quantum
          authorauthorM. Horowitz and authorE. Grumbling, @noop   (year2019)NoStop
         [Schuman et al.(2017)Schuman, Potok, Patton, Birdwell, Dean, Rose, and Plank]schuman2017survey
          authorauthorC. D. Schuman, authorT. E. Potok,
          authorR. M. Patton, authorJ. D. Birdwell, authorM. E. Dean, authorG. S. Rose,  and authorJ. S. Plank, @noop journaljournalarXiv preprint arXiv:1705.06963  (year2017)NoStop
         [Sun et al.(2020)Sun,
          Song, Hong, and Li]sun2020review
          authorauthorC. Sun, authorM. Song, authorS. Hong,  and authorH. Li, @noop journaljournalarXiv preprint arXiv:2012.02974  (year2020)NoStop
         [MacLennan(2007)]maclennan2007review
          authorauthorB. J. MacLennan, @noop journaljournalDepartment of Electrical Engineering & Computer Science, University of
          Tennessee, Technical Report UT-CS-07-601 (September) , pages19798 (year2007)NoStop
         [Ulmann(2013)]ulmann2013analog
          authorauthorB. Ulmann, @noop titleAnalog computing  (publisherOldenbourg Wissenschaftsverlag Verlag, year2013)NoStop
         [Csaba and Porod(2020)]csaba2020coupled
          authorauthorG. Csaba and authorW. Porod,@noop journaljournalApplied physics
          reviews volume7 (year2020)NoStop
         [Conte et al.(2019)Conte,
          DeBenedictis, Ganesh, Hylton,
          Strachan, Williams, Alemi,
          Altenberg, Crooks, Crutchfieldet al.]conte2019thermodynamic
          authorauthorT. Conte, authorE. DeBenedictis,
          authorN. Ganesh, authorT. Hylton, authorJ. P. Strachan, authorR. S. Williams, authorA. Alemi, authorL. Altenberg, authorG. Crooks, authorJ. Crutchfield,  et al., @noop journaljournalarXiv preprint arXiv:1911.01968 
          (year2019)NoStop
         [Hylton(2020)]hylton2020thermodynamic
          authorauthorT. Hylton, in @noop booktitleProceedings,  Vol. volume47 (organizationMDPI, year2020) p. pages23NoStop
         [Aifer et al.(2023)Aifer,
          Donatella, Gordon, Ahle,
          Simpson, Crooks, and Coles]aifer2023thermodynamic
          authorauthorM. Aifer, authorK. Donatella,
          authorM. H. Gordon, authorT. Ahle, authorD. Simpson, authorG. E. Crooks,  and authorP. J. Coles, @noop journaljournalarXiv preprint arXiv:2308.05660  (year2023)NoStop
         [Von Neumann(1957)]von1957non
          authorauthorJ. Von Neumann, @noop titleNon-linear
          capacitance or inductance switching, amplifying, and memory organs, 
          (year1957), noteUS Patent
          2,815,488NoStop
         [Ciliberto(2017)]ciliberto2017experiments
          authorauthorS. Ciliberto, @noop journaljournalPhysical Review X volume7, pages021051 (year2017)NoStop
         [Dago et al.(2021)Dago,
          Pereda, Barros, Ciliberto,  and Bellon]dago2021information
          authorauthorS. Dago, authorJ. Pereda,
          authorN. Barros, authorS. Ciliberto,  and authorL. Bellon, @noop journaljournalPhysical Review Lettersvolume126, pages170601 (year2021)NoStop
         [Dago et al.(2022)Dago,
          Pereda, Ciliberto, and Bellon]dago2022virtual
          authorauthorS. Dago, authorJ. Pereda,
          authorS. Ciliberto,  andauthorL. Bellon, @noop journaljournalJournal of Statistical Mechanics:
          Theory and Experiment volume2022, pages053209 (year2022)NoStop
         [Wang and Roychowdhury(2019)]wang2019oim
          authorauthorT. Wang and authorJ. Roychowdhury, in @noop booktitleUnconventional Computation and Natural Computation: 18th International
          Conference, UCNC 2019, Tokyo, Japan, June 3–7, 2019, Proceedings 18  (organizationSpringer, year2019) pp. pages232–256NoStop
         [Chou et al.(2019)Chou,
          Bramhavar, Ghosh, and Herzog]chou2019analog
          authorauthorJ. Chou, authorS. Bramhavar,
          authorS. Ghosh,  and authorW. Herzog, @noop journaljournalScientific Reports volume9, pages14786 (year2019)NoStop
         [Melanson et al.(2023)Melanson, Khater, Aifer, Donatella, Gordon, Ahle, Crooks, Martinez, Sbahi, andColes]melanson2023thermodynamic
          authorauthorD. Melanson, authorM. A. Khater, authorM. Aifer,
          authorK. Donatella, authorM. H. Gordon, authorT. Ahle, authorG. Crooks, authorA. J.Martinez, authorF. Sbahi,  and authorP. J. Coles, @noop journaljournalarXiv
          preprint arXiv:2312.04836  (year2023)NoStop
         [Goto(1959)]goto1959parametron
          authorauthorE. Goto, @noop journaljournalProceedings of the IRE volume47, pages1304 (year1959)NoStop
         [Bonnin et al.(2022)Bonnin,
          Traversa, and Bonani]bonnin2022coupled
          authorauthorM. Bonnin, authorF. L. Traversa,  and authorF. Bonani, @noop journaljournalAdvances in Artificial Intelligence-based Technologies: Selected Papers in
          Honour of Professor Nikolaos G. Bourbakis-Vol. 1 , pages179
          (year2022)NoStop
         [Rumelhart et al.(1986)Rumelhart, Hinton, and Williams]rumelhart1986learning
          authorauthorD. E. Rumelhart, authorG. E. Hinton,  and authorR. J. Williams, @noop journaljournalBiometrika volume71, pages599
          (year1986)NoStop
         [Gidon et al.(2020)Gidon,
          Zolnik, Fidzinski, Bolduan,
          Papoutsi, Poirazi, Holtkamp,
          Vida, and Larkum]gidon2020dendritic
          authorauthorA. Gidon, authorT. A. Zolnik,
          authorP. Fidzinski, authorF. Bolduan, authorA. Papoutsi, authorP. Poirazi, authorM. Holtkamp, authorI. Vida,  and authorM. E.Larkum, @noop journaljournalScience volume367, pages83 (year2020)NoStop
         [Noel et al.(2021)Noel,
          Bharadwaj, Muthiah-Nakarajan, Dutta, and Amali]noel2021biologically
          authorauthorM. M. Noel, authorS. Bharadwaj,
          authorV. Muthiah-Nakarajan,
          authorP. Dutta,  and authorG. B. Amali, @noop journaljournalarXiv preprint arXiv:2111.04020 (year2021)NoStop
         [Effenberger et al.(2022)Effenberger, Carvalho, Dubinin, andSinger]effenberger2022functional
          authorauthorF. Effenberger, authorP. Carvalho, authorI. Dubinin,   and authorW. Singer,@noop journaljournalbioRxiv ,pages2022 (year2022)NoStop
         [Landauer(1961)]landauer1961irreversibility
          authorauthorR. Landauer, @noop journaljournalIBM
          journal of research and development volume5,pages183 (year1961)NoStop
         [Sagawa(2012)]sagawa2012thermodynamics
          authorauthorT. Sagawa, @noop journaljournalProgress of theoretical physics volume127,pages1 (year2012)NoStop
         [Zulkowski and DeWeese(2014)]zulkowski2014optimal
          authorauthorP. R. Zulkowski and authorM. R. DeWeese, @noop journaljournalPhysical Review E volume89, pages052140 (year2014)NoStop
         [Proesmans et al.(2020)Proesmans, Ehrich, and Bechhoefer]proesmans2020finite
          authorauthorK. Proesmans, authorJ. Ehrich,   and authorJ. Bechhoefer,@noop journaljournalPhysical Review
          Letters volume125, pages100602
          (year2020)NoStop
         [Dago and Bellon(2022)]dago2022dynamics
          authorauthorS. Dago and authorL. Bellon,@noop journaljournalPhysical Review
          Letters volume128, pages070604
          (year2022)NoStop
         [Note1()]Note1
          noteSingle operations and multiple operations incur the same
          energy cost, with the caveat that the input I to the neuron must be
          maintained long enough to observe these operations.Stop
         [LeCun et al.(1998)LeCun,
          Bottou, Bengio, and Haffner]lecun1998gradient
          authorauthorY. LeCun, authorL. Bottou,
          authorY. Bengio,  and authorP. Haffner, @noop journaljournalProceedings of the IEEEvolume86, pages2278 (year1998)NoStop
         [Note2()]Note2
          noteIt could happen during training that Ω _α ^2
          becomes negative for certain α, indicating overdamping, in which case
          we must make the replacement Ω→ i Ω in (<ref>) and (<ref>). This did not happen in the simulations reported here.Stop
