Geometric Bound on the Efficiency of Irreversible Thermodynamic Cycles.

Stochastic thermodynamics has revolutionized our understanding of heat engines operating in finite time. Recently, numerous studies have considered the optimal operation of thermodynamic cycles acting as heat engines with a given profile in thermodynamic space (e.g., P-V space in classical thermodynamics), with a particular focus on the Carnot engine. In this work, we use the lens of thermodynamic geometry to explore the full space of thermodynamic cycles with continuously varying bath temperature in search of optimally shaped cycles acting in the slow-driving regime. We apply classical isoperimetric inequalities to derive a universal geometric bound on the efficiency of any irreversible thermodynamic cycle and explicitly construct efficient heat engines operating in finite time that nearly saturate this bound for a specific model system. Given the bound, these optimal cycles perform more efficiently than all other thermodynamic cycles operating as heat engines in finite time, including notable cycles, such as those of Carnot, Stirling, and Otto. For example, in comparison to recent experiments, this corresponds to orders of magnitude improvement in the efficiency of engines operating in certain time regimes. Our results suggest novel design principles for future mesoscopic heat engines and are ripe for experimental investigation.

Critical Point-Finding Methods Reveal Gradient-Flat Regions of Deep Network Losses.

Despite the fact that the loss functions of deep neural networks are highly nonconvex, gradient-based optimization algorithms converge to approximately the same performance from many random initial points. One thread of work has focused on explaining this phenomenon by numerically characterizing the local curvature near critical points of the loss function, where the gradients are near zero. Such studies have reported that neural network losses enjoy a no-bad-local-minima property, in disagreement with more recent theoretical results. We report here that the methods used to find these putative critical points suffer from a bad local minima problem of their own: they often converge to or pass through regions where the gradient norm has a stationary point. We call these gradient-flat regions, since they arise when the gradient is approximately in the kernel of the Hessian, such that the loss is locally approximately linear, or flat, in the direction of the gradient. We describe how the presence of these regions necessitates care in both interpreting past results that claimed to find critical points of neural network losses and in designing second-order methods for optimizing neural networks.

Efficient sensory coding of multidimensional stimuli.

According to the efficient coding hypothesis, sensory systems are adapted to maximize their ability to encode information about the environment. Sensory neurons play a key role in encoding by selectively modulating their firing rate for a subset of all possible stimuli. This pattern of modulation is often summarized via a tuning curve. The optimally efficient distribution of tuning curves has been calculated in variety of ways for one-dimensional (1-D) stimuli. However, many sensory neurons encode multiple stimulus dimensions simultaneously. It remains unclear how applicable existing models of 1-D tuning curves are for neurons tuned across multiple dimensions. We describe a mathematical generalization that builds on prior work in 1-D to predict optimally efficient multidimensional tuning curves. Our results have implications for interpreting observed properties of neuronal populations. For example, our results suggest that not all tuning curve attributes (such as gain and bandwidth) are equally useful for evaluating the encoding efficiency of a population.

Heterogeneous Synaptic Weighting Improves Neural Coding in the Presence of Common Noise.

Simultaneous recordings from the cortex have revealed that neural activity is highly variable and that some variability is shared across neurons in a population. Further experimental work has demonstrated that the shared component of a neuronal population's variability is typically comparable to or larger than its private component. Meanwhile, an abundance of theoretical work has assessed the impact that shared variability has on a population code. For example, shared input noise is understood to have a detrimental impact on a neural population's coding fidelity. However, other contributions to variability, such as common noise, can also play a role in shaping correlated variability. We present a network of linear-nonlinear neurons in which we introduce a common noise input to model-for instance, variability resulting from upstream action potentials that are irrelevant to the task at hand. We show that by applying a heterogeneous set of synaptic weights to the neural inputs carrying the common noise, the network can improve its coding ability as measured by both Fisher information and Shannon mutual information, even in cases where this results in amplification of the common noise. With a broad and heterogeneous distribution of synaptic weights, a population of neurons can remove the harmful effects imposed by afferents that are uninformative about a stimulus. We demonstrate that some nonlinear networks benefit from weight diversification up to a certain population size, above which the drawbacks from amplified noise dominate over the benefits of diversification. We further characterize these benefits in terms of the relative strength of shared and private variability sources. Finally, we studied the asymptotic behavior of the mutual information and Fisher information analytically in our various networks as a function of population size. We find some surprising qualitative changes in the asymptotic behavior as we make seemingly minor changes in the synaptic weight distributions.

Design of optical neural networks with component imprecisions.

For the benefit of designing scalable, fault resistant optical neural networks (ONNs), we investigate the effects architectural designs have on the ONNs' robustness to imprecise components. We train two ONNs - one with a more tunable design (GridNet) and one with better fault tolerance (FFTNet) - to classify handwritten digits. When simulated without any imperfections, GridNet yields a better accuracy (∼98%) than FFTNet (∼95%). However, under a small amount of error in their photonic components, the more fault tolerant FFTNet overtakes GridNet. We further provide thorough quantitative and qualitative analyses of ONNs' sensitivity to varying levels and types of imprecisions. Our results offer guidelines for the principled design of fault-tolerant ONNs as well as a foundation for further research.

Entropy Production and Fluctuation Theorems for Active Matter.

Active biological systems reside far from equilibrium, dissipating heat even in their steady state, thus requiring an extension of conventional equilibrium thermodynamics and statistical mechanics. In this Letter, we have extended the emerging framework of stochastic thermodynamics to active matter. In particular, for the active Ornstein-Uhlenbeck model, we have provided consistent definitions of thermodynamic quantities such as work, energy, heat, entropy, and entropy production at the level of single, stochastic trajectories and derived related fluctuation relations. We have developed a generalization of the Clausius inequality, which is valid even in the presence of the non-Hamiltonian dynamics underlying active matter systems. We have illustrated our results with explicit numerical studies.

Minimum and Maximum Entropy Distributions for Binary Systems with Known Means and Pairwise Correlations.

Maximum entropy models are increasingly being used to describe the collective activity of neural populations with measured mean neural activities and pairwise correlations, but the full space of probability distributions consistent with these constraints has not been explored. We provide upper and lower bounds on the entropy for the minimum entropy distribution over arbitrarily large collections of binary units with any fixed set of mean values and pairwise correlations. We also construct specific low-entropy distributions for several relevant cases. Surprisingly, the minimum entropy solution has entropy scaling logarithmically with system size for any set of first- and second-order statistics consistent with arbitrarily large systems. We further demonstrate that some sets of these low-order statistics can only be realized by small systems. Our results show how only small amounts of randomness are needed to mimic low-order statistical properties of highly entropic distributions, and we discuss some applications for engineered and biological information transmission systems.

Inhibitory interneurons decorrelate excitatory cells to drive sparse code formation in a spiking model of V1.

Sparse coding models of natural scenes can account for several physiological properties of primary visual cortex (V1), including the shapes of simple cell receptive fields (RFs) and the highly kurtotic firing rates of V1 neurons. Current spiking network models of pattern learning and sparse coding require direct inhibitory connections between the excitatory simple cells, in conflict with the physiological distinction between excitatory (glutamatergic) and inhibitory (GABAergic) neurons (Dale's Law). At the same time, the computational role of inhibitory neurons in cortical microcircuit function has yet to be fully explained. Here we show that adding a separate population of inhibitory neurons to a spiking model of V1 provides conformance to Dale's Law, proposes a computational role for at least one class of interneurons, and accounts for certain observed physiological properties in V1. When trained on natural images, this excitatory-inhibitory spiking circuit learns a sparse code with Gabor-like RFs as found in V1 using only local synaptic plasticity rules. The inhibitory neurons enable sparse code formation by suppressing predictable spikes, which actively decorrelates the excitatory population. The model predicts that only a small number of inhibitory cells is required relative to excitatory cells and that excitatory and inhibitory input should be correlated, in agreement with experimental findings in visual cortex. We also introduce a novel local learning rule that measures stimulus-dependent correlations between neurons to support "explaining away" mechanisms in neural coding.

Up states are rare in awake auditory cortex.

The dynamics of subthreshold membrane potential provide insight into the organization of activity in neural circuits. In many brain areas, membrane potential is bistable, transiting between a relatively hyperpolarized down state and a depolarized up state. These up and down states, which have been proposed to play a number of computational roles, have mainly been studied in anesthetized and in vitro preparations. Here, we have used intracellular recordings to characterize the dynamics of membrane potential in the auditory cortex of awake rats. We find that long up states are rare in the awake auditory cortex, with only 0.4% of up states >500 ms. Most neurons displayed only brief up states (bumps) and spent on average ∼1% of recording time in up states >500 ms. We suggest that the near absence of long up states in awake auditory cortex may reflect an adaptation to the rapid processing of auditory stimuli.

Early selection of task-relevant features through population gating.

Brains can gracefully weed out irrelevant stimuli to guide behavior. This feat is believed to rely on a progressive selection of task-relevant stimuli across the cortical hierarchy, but the specific across-area interactions enabling stimulus selection are still unclear. Here, we propose that population gating, occurring within primary auditory cortex (A1) but controlled by top-down inputs from prelimbic region of medial prefrontal cortex (mPFC), can support across-area stimulus selection. Examining single-unit activity recorded while rats performed an auditory context-dependent task, we found that A1 encoded relevant and irrelevant stimuli along a common dimension of its neural space. Yet, the relevant stimulus encoding was enhanced along an extra dimension. In turn, mPFC encoded only the stimulus relevant to the ongoing context. To identify candidate mechanisms for stimulus selection within A1, we reverse-engineered low-rank RNNs trained on a similar task. Our analyses predicted that two context-modulated neural populations gated their preferred stimulus in opposite contexts, which we confirmed in further analyses of A1. Finally, we show in a two-region RNN how population gating within A1 could be controlled by top-down inputs from PFC, enabling flexible across-area communication despite fixed inter-areal connectivity.

Limited-control optimal protocols arbitrarily far from equilibrium.

Recent studies have explored finite-time dissipation-minimizing protocols for stochastic thermodynamic systems driven arbitrarily far from equilibrium, when granted full external control to drive the system. However, in both simulation and experimental contexts, systems often may only be controlled with a limited set of degrees of freedom. Here, going beyond slow- and fast-driving approximations employed in previous studies, we obtain exact finite-time optimal protocols for this limited-control setting. By working with deterministic Fokker-Planck probability density time evolution, we can frame the work-minimizing protocol problem in the standard form of an optimal control theory problem. We demonstrate that finding the exact optimal protocol is equivalent to solving a system of Hamiltonian partial differential equations, which in many cases admit efficiently calculable numerical solutions. Within this framework, we reproduce analytical results for the optimal control of harmonic potentials and numerically devise optimal protocols for two anharmonic examples: varying the stiffness of a quartic potential and linearly biasing a double-well potential. We confirm that these optimal protocols outperform other protocols produced through previous methods, in some cases by a substantial amount. We find that for the linearly biased double-well problem, the mean position under the optimal protocol travels at a near-constant velocity. Surprisingly, for a certain timescale and barrier height regime, the optimal protocol is also nonmonotonic in time.

Optimal finite-time Brownian Carnot engine.

Recent advances in experimental control of colloidal systems have spurred a revolution in the production of mesoscale thermodynamic devices. Functional "textbook" engines, such as the Stirling and Carnot cycles, have been produced in colloidal systems where they operate far from equilibrium. Simultaneously, significant theoretical advances have been made in the design and analysis of such devices. Here, we use methods from thermodynamic geometry to characterize the optimal finite-time nonequilibrium cyclic operation of the parametric harmonic oscillator in contact with a time-varying heat bath with particular focus on the Brownian Carnot cycle. We derive the optimally parametrized Carnot cycle, along with two other new cycles and compare their dissipated energy, efficiency, and steady-state power production against each other and a previously tested experimental protocol for the Carnot cycle. We demonstrate a 20% improvement in dissipated energy over previous experimentally tested protocols and a ∼50% improvement under other conditions for one of our engines, whereas our final engine is more efficient and powerful than the others we considered. Our results provide the means for experimentally realizing optimal mesoscale heat engines.

Solution to the Fokker-Planck equation for slowly driven Brownian motion: Emergent geometry and a formula for the corresponding thermodynamic metric.

Considerable progress has recently been made with geometrical approaches to understanding and controlling small out-of-equilibrium systems, but a mathematically rigorous foundation for these methods has been lacking. Towards this end, we develop a perturbative solution to the Fokker-Planck equation for one-dimensional driven Brownian motion in the overdamped limit enabled by the spectral properties of the corresponding single-particle Schrödinger operator. The perturbation theory is in powers of the inverse characteristic timescale of variation of the fastest varying control parameter, measured in units of the system timescale, which is set by the smallest eigenvalue of the corresponding Schrödinger operator. It applies to any Brownian system for which the Schrödinger operator has a confining potential. We use the theory to rigorously derive an exact formula for a Riemannian "thermodynamic" metric in the space of control parameters of the system. We show that up to second-order terms in the perturbation theory, optimal dissipation-minimizing driving protocols minimize the length defined by this metric. We also show that a previously proposed metric is calculable from our exact formula with corrections that are exponentially suppressed in a characteristic length scale. We illustrate our formula using the two-dimensional example of a harmonic oscillator with time-dependent spring constant in a time-dependent electric field. Lastly, we demonstrate that the Riemannian geometric structure of the optimal control problem is emergent; it derives from the form of the perturbative expansion for the probability density and persists to all orders of the expansion.

New method for parameter estimation in probabilistic models: minimum probability flow.

Fitting probabilistic models to data is often difficult, due to the general intractability of the partition function. We propose a new parameter fitting method, minimum probability flow (MPF), which is applicable to any parametric model. We demonstrate parameter estimation using MPF in two cases: a continuous state space model, and an Ising spin glass. In the latter case, MPF outperforms current techniques by at least an order of magnitude in convergence time with lower error in the recovered coupling parameters.

Sparse representation of sounds in the unanesthetized auditory cortex.

How do neuronal populations in the auditory cortex represent acoustic stimuli? Although sound-evoked neural responses in the anesthetized auditory cortex are mainly transient, recent experiments in the unanesthetized preparation have emphasized subpopulations with other response properties. To quantify the relative contributions of these different subpopulations in the awake preparation, we have estimated the representation of sounds across the neuronal population using a representative ensemble of stimuli. We used cell-attached recording with a glass electrode, a method for which single-unit isolation does not depend on neuronal activity, to quantify the fraction of neurons engaged by acoustic stimuli (tones, frequency modulated sweeps, white-noise bursts, and natural stimuli) in the primary auditory cortex of awake head-fixed rats. We find that the population response is sparse, with stimuli typically eliciting high firing rates (>20 spikes/second) in less than 5% of neurons at any instant. Some neurons had very low spontaneous firing rates (<0.01 spikes/second). At the other extreme, some neurons had driven rates in excess of 50 spikes/second. Interestingly, the overall population response was well described by a lognormal distribution, rather than the exponential distribution that is often reported. Our results represent, to our knowledge, the first quantitative evidence for sparse representations of sounds in the unanesthetized auditory cortex. Our results are compatible with a model in which most neurons are silent much of the time, and in which representations are composed of small dynamic subsets of highly active neurons.

Engineered swift equilibration for arbitrary geometries.

Engineered swift equilibration (ESE) is a class of driving protocols that enforce an equilibrium distribution with respect to external control parameters at the beginning and end of rapid state transformations of open, classical nonequilibrium systems. ESE protocols have previously been derived and experimentally realized for Brownian particles in simple, one-dimensional, time-varying trapping potentials; one recent study considered ESE in two-dimensional Euclidean configuration space. Here we extend the ESE framework to generic, overdamped Brownian systems in arbitrary curved configuration space and illustrate our results with specific examples not amenable to previous techniques. Our approach may be used to impose the necessary dynamics to control the full temporal configurational distribution in a wide variety of experimentally realizable settings.

Reliability and representational bandwidth in the auditory cortex.

It is unclear why there are so many more neurons in sensory cortex than in the sensory periphery. One possibility is that these "extra" neurons are used to overcome cortical noise and faithfully represent the acoustic stimulus. Another possibility is that even after overcoming cortical noise, there is "excess representational bandwidth" available and that this bandwidth is used to represent conjunctions of auditory and nonauditory information for computation. Here, we discuss recent data about neuronal reliability in auditory cortex showing that cortical noise may not be as high as was previously believed. Although at present, the data suggest that auditory cortex neurons can be more reliable than those in the visual cortex, we speculate that the principles governing cortical computation are universal and that visual and other cortical areas can also exploit strategies based on similarly high-fidelity activity.

Neural gallops across auditory streams.

We continually rely on our ability to segregate the myriad sounds in our environment--phones ringing, people talking--into separate "auditory streams," each originating from a different source. In this issue of Neuron, Micheyl et al. provide the most direct evidence to date linking single-unit spiking responses from auditory cortex to the perception of distinct auditory streams.