Gaussian processes are now commonly used in dimensionality reduction approaches tailored to neuroscience, especially to describe changes in high-dimensional neural activity over time. As recording capabilities expand to include neuronal populations across multiple brain areas, cortical layers, and cell types, interest in extending gaussian process factor models to characterize multipopulation interactions has grown. However, the cubic runtime scaling of current methods with the length of experimental trials and the number of recorded populations (groups) precludes their application to large-scale multipopulation recordings. Here, we improve this scaling from cubic to linear in both trial length and group number. We present two approximate approaches to fitting multigroup gaussian process factor models based on inducing variables and the frequency domain. Empirically, both methods achieved orders of magnitude speed-up with minimal impact on statistical performance, in simulation and on neural recordings of hundreds of neurons across three brain areas. The frequency domain approach, in particular, consistently provided the greatest runtime benefits with the fewest trade-offs in statistical performance. We further characterize the estimation biases introduced by the frequency domain approach and demonstrate effective strategies to mitigate them. This work enables a powerful class of analysis techniques to keep pace with the growing scale of multipopulation recordings, opening new avenues for exploring brain function.

Hippocampal remapping, in which place cells form distinct activity maps across different environments, is a well-established phenomenon with a range of theoretical interpretations. Some theories propose that remapping helps to minimize interference between competing spatial memories, whereas others link it to shifts in an underlying latent state representation. However, how these interpretations of remapping relate to one another, and what types of activity changes they are compatible with, remains unclear. To unify and elucidate the mechanisms behind remapping, we here adopt a neural coding and population geometry perspective. Assuming that hippocampal population activity can be understood through a linearly-decodable latent space, we show that there are three possible mechanisms to induce remapping: (i) a true change in the mapping between neural and latent space, (ii) modulation of activity due to non-spatial mixed selectivity of place cells, or (iii) neural variability in the null space of the latent space that reflects a redundant code. We simulate and visualize examples of these remapping types in a network model, and relate the resultant remapping behavior to various models and experimental findings in the literature. Overall, our work serves as a unifying framework with which to visualize, understand, and compare the wide array of theories and experimental observations about remapping, and may serve as a testbed for understanding neural response variability under various experimental conditions.

The geometrical and statistical properties of brain activity depend on the way neurons connect to form recurrent circuits. However, the link between connectivity structure and emergent activity remains incompletely understood. We investigate this relationship in recurrent neural networks with additive stochastic inputs. We assume that the synaptic connectivity can be expressed in a low-rank form, parameterized by a handful of connectivity vectors, and examine how the geometry of emergent activity relates to these vectors. Our findings reveal that this relationship critically depends on the dimensionality of the external stochastic inputs. When inputs are low-dimensional, activity remains low-dimensional, and recurrent dynamics influence it within a subspace spanned by a subset of the connectivity vectors, with dimensionality equal to the rank of the connectivity matrix. In contrast, when inputs are high-dimensional, activity also becomes potentially high-dimensional. The contribution of recurrent dynamics is apparent within a subspace spanned by the totality of the connectivity vectors, with dimensionality equal to twice the rank of the connectivity matrix. Applying our formalism to excitatory-inhibitory networks, we discuss how the input configuration also plays a crucial role in determining the amount of amplification generated by non-normal dynamics. Our work provides a foundation for studying activity in structured brain circuits under realistic noise conditions, and offers a framework for interpreting stochastic models inferred from experimental data.

A central objective in neuroscience is to understand how the brain orchestrates movement. Recent advances in automated tracking technologies have made it possible to document behavior with unprecedented temporal resolution and scale, generating rich datasets which can be exploited to gain insights into the neural control of movement. One common approach is to identify stereotypical motor primitives using cluster analysis. However, this categorical description can limit our ability to model the effect of more continuous control schemes. Here we take a control theoretic approach to behavioral modeling and argue that movements can be understood as the output of a controlled dynamical system. Previously, models of movement dynamics, trained solely on behavioral data, have been effective in reproducing observed features of neural activity. These models addressed specific scenarios where animals were trained to execute particular movements upon receiving a prompt. In this study, we extend this approach to analyze the full natural locomotor repertoire of an animal: the zebrafish larva. Our findings demonstrate that this repertoire can be effectively generated through a sparse control signal driving a latent Recurrent Neural Network (RNN). Our model's learned latent space preserves key kinematic features and disentangles different categories of movements. To further interpret the latent dynamics, we used balanced model reduction to yield a simplified model. Collectively, our methods serve as a case study for interpretable system identification, and offer a novel framework for understanding neural activity in relation to movement.

Deep feedforward and recurrent neural networks have become successful functional models of the brain, but they neglect obvious biological details such as spikes and Dale's law. Here we argue that these details are crucial in order to understand how real neural circuits operate. Towards this aim, we put forth a new framework for spike-based computation in low-rank excitatory-inhibitory spiking networks. By considering populations with rank-1 connectivity, we cast each neuron's spiking threshold as a boundary in a low-dimensional input-output space. We then show how the combined thresholds of a population of inhibitory neurons form a stable boundary in this space, and those of a population of excitatory neurons form an unstable boundary. Combining the two boundaries results in a rank-2 excitatory-inhibitory (EI) network with inhibition-stabilized dynamics at the intersection of the two boundaries. The computation of the resulting networks can be understood as the difference of two convex functions and is thereby capable of approximating arbitrary non-linear input-output mappings. We demonstrate several properties of these networks, including noise suppression and amplification, irregular activity and synaptic balance, as well as how they relate to rate network dynamics in the limit that the boundary becomes soft. Finally, while our work focuses on small networks (5-50 neurons), we discuss potential avenues for scaling up to much larger networks. Overall, our work proposes a new perspective on spiking networks that may serve as a starting point for a mechanistic understanding of biological spike-based computation.

Modern recording techniques now allow us to record from distinct neuronal populations in different brain networks. However, especially as we consider multiple (more than two) populations, new conceptual and statistical frameworks are needed to characterize the multi-dimensional, concurrent flow of signals among these populations. Here, we develop a dimensionality reduction framework that determines (1) the subset of populations described by each latent dimension, (2) the direction of signal flow among those populations, and (3) how those signals evolve over time within and across experimental trials. We illustrate these features in simulation, and further validate the method by applying it to previously studied recordings from neuronal populations in macaque visual areas V1 and V2. Then we study interactions across select laminar compartments of areas V1, V2, and V3d, recorded simultaneously with multiple Neuropixels probes. Our approach uncovered signatures of selective communication across these three areas that related to their retinotopic alignment. This work advances the study of concurrent signaling across multiple neuronal populations.

Higher-order brain areas such as the frontal cortices are considered essential for the flexible solution of tasks. However, the precise computational role of these areas is still debated. Indeed, even for the simplest of tasks, we cannot really explain how the measured brain activity, which evolves over time in complicated ways, relates to the task structure. Here, we follow a normative approach, based on integrating the principle of efficient coding with the framework of Markov decision processes (MDP). More specifically, we focus on MDPs whose state is based on action-observation histories, and we show how to compress the state space such that unnecessary redundancy is eliminated, while task-relevant information is preserved. We show that the efficiency of a state space representation depends on the (long-term) behavioural goal of the agent, and we distinguish between model-based and habitual agents. We apply our approach to simple tasks that require short-term memory, and we show that the efficient state space representations reproduce the key dynamical features of recorded neural activity in frontal areas (such as ramping, sequentiality, persistence). If we additionally assume that neural systems are subject to accuracy-cost tradeoffs, we find a surprising match to neural data on a population level.

Neurons mainly communicate through spikes, and much effort has been spent to understand how the dynamics of spiking neural networks (SNNs) relates to their connectivity. Meanwhile, most major advances in machine learning have been made with simpler, rate-based networks, with SNNs only recently showing competitive results, largely thanks to transferring insights from rate to spiking networks. However, it is still an open question exactly which computations SNNs perform. Recently, the time-averaged firing rates of several SNNs were shown to yield the solutions to convex optimization problems. Here we turn these findings around and show that virtually all inhibition-dominated SNNs can be understood through the lens of convex optimization, with network connectivity, timescales, and firing thresholds being intricately linked to the parameters of underlying convex optimization problems. This approach yields new, geometric insights into the computations performed by spiking networks. In particular, we establish a class of SNNs whose instantaneous output provides a solution to linear or quadratic programming problems, and we thereby reveal their input-output mapping. Using these insights, we derive local, supervised learning rules that can approximate given convex input-output functions, and we show that the resulting networks are consistent with many features from biological networks, such as low firing rates, irregular firing, E/I balance, and robustness to perturbations and synaptic delays.

Learning depends on changes in synaptic connections deep inside the brain. In multilayer networks, these changes are triggered by error signals fed back from the output, generally through a stepwise inversion of the feedforward processing steps. The gold standard for this process --- backpropagation --- works well in artificial neural networks, but is biologically implausible. Several recent proposals have emerged to address this problem, but many of these biologically-plausible schemes are based on learning an independent set of feedback connections. This complicates the assignment of errors to each synapse by making it dependent upon a second learning problem, and by fitting inversions rather than guaranteeing them. Here, we show that feedforward network transformations can be effectively inverted through dynamics. We derive this dynamic inversion from the perspective of feedback control, where the forward transformation is reused and dynamically interacts with fixed or random feedback to propagate error signals during the backward pass. Importantly, this scheme does not rely upon a second learning problem for feedback because accurate inversion is guaranteed through the network dynamics. We map these dynamics onto generic feedforward networks, and show that the resulting algorithm performs well on several supervised and unsupervised datasets. Finally, we discuss potential links between dynamic inversion and second-order optimization. Overall, our work introduces an alternative perspective on credit assignment in the brain, and proposes a special role for temporal dynamics and feedback control during learning.

Networks based on coordinated spike coding can encode information with high efficiency in the spike trains of individual neurons. These networks exhibit single-neuron variability and tuning curves as typically observed in cortex, but paradoxically coincide with a precise, non-redundant spike-based population code. However, it has remained unclear whether the specific synaptic connectivities required in these networks can be learnt with local learning rules. Here, we show how to learn the required architecture. Using coding efficiency as an objective, we derive spike-timing-dependent learning rules for a recurrent neural network, and we provide exact solutions for the networks' convergence to an optimal state. As a result, we deduce an entire network from its input distribution and a firing cost. After learning, basic biophysical quantities such as voltages, firing thresholds, excitation, inhibition, or spikes acquire precise functional interpretations.

The brain uses spikes in neural circuits to perform many dynamical computations. The computations are performed with properties such as spiking efficiency, i.e. minimal number of spikes, and robustness to noise. A major obstacle for learning computations in artificial spiking neural networks with such desired biological properties is due to lack of our understanding of how biological spiking neural networks learn computations. Here, we consider the credit assignment problem, i.e. determining the local contribution of each synapse to the network's global output error, for learning nonlinear dynamical computations in a spiking network with the desired properties of biological networks. We approach this problem by fusing the theory of efficient, balanced neural networks (EBN) with nonlinear adaptive control theory to propose a local learning rule. Locality of learning rules are ensured by feeding back into the network its own error, resulting in a learning rule depending solely on presynaptic inputs and error feedbacks. The spiking efficiency and robustness of the network are guaranteed by maintaining a tight excitatory/inhibitory balance, ensuring that each spike represents a local projection of the global output error and minimizes a loss function. The resulting networks can learn to implement complex dynamics with very small numbers of neurons and spikes, exhibit the same spike train variability as observed experimentally, and are extremely robust to noise and neuronal loss.

All of our perceptual experiences arise from the activity of neural populations. Here we study the formation of such percepts under the assumption that they emerge from a linear readout, i.e., a weighted sum of the neurons' firing rates. We show that this assumption constrains the trial-to-trial covariance structure of neural activities and animal behavior. The predicted covariance structure depends on the readout parameters, and in particular on the temporal integration window w and typical number of neurons K used in the formation of the percept. Using these predictions, we show how to infer the readout parameters from joint measurements of a subject's behavior and neural activities. We consider three such scenarios: (1) recordings from the complete neural population, (2) recordings of neuronal sub-ensembles whose size exceeds K, and (3) recordings of neuronal sub-ensembles that are smaller than K. Using theoretical arguments and artificially generated data, we show that the first two scenarios allow us to recover the typical spatial and temporal scales of the readout. In the third scenario, we show that the readout parameters can only be recovered by making additional assumptions about the structure of the full population activity. Our work provides the first thorough interpretation of (feed-forward) percept formation from a population of sensory neurons. We discuss applications to experimental recordings in classic sensory decision-making tasks, which will hopefully provide new insights into the nature of perceptual integration.

How are firing rates in a spiking network related to neural input, connectivity and network function? This is an important problem because firing rates are one of the most important measures of network activity, in both the study of neural computation and neural network dynamics. However, it is a difficult problem, because the spiking mechanism of individual neurons is highly non-linear, and these individual neurons interact strongly through connectivity. We develop a new technique for calculating firing rates in optimal balanced networks. These are particularly interesting networks because they provide an optimal spike-based signal representation while producing cortex-like spiking activity through a dynamic balance of excitation and inhibition. We can calculate firing rates by treating balanced network dynamics as an algorithm for optimizing signal representation. We identify this algorithm and then calculate firing rates by finding the solution to the algorithm. Our firing rate calculation relates network firing rates directly to network input, connectivity and function. This allows us to explain the function and underlying mechanism of tuning curves in a variety of systems.

Two observations about the cortex have puzzled neuroscientists for a long time. First, neural responses are highly variable. Second, the level of excitation and inhibition received by each neuron is tightly balanced at all times. Here, we demonstrate that both properties are necessary consequences of neural networks that represent information efficiently in their spikes. We illustrate this insight with spiking networks that represent dynamical variables. Our approach is based on two assumptions: We assume that information about dynamical variables can be read out linearly from neural spike trains, and we assume that neurons only fire a spike if that improves the representation of the dynamical variables. Based on these assumptions, we derive a network of leaky integrate-and-fire neurons that is able to implement arbitrary linear dynamical systems. We show that the membrane voltage of the neurons is equivalent to a prediction error about a common population-level signal. Among other things, our approach allows us to construct an integrator network of spiking neurons that is robust against many perturbations. Most importantly, neural variability in our networks cannot be equated to noise. Despite exhibiting the same single unit properties as widely used population code models (e.g. tuning curves, Poisson distributed spike trains), balanced networks are orders of magnitudes more reliable. Our approach suggests that spikes do matter when considering how the brain computes, and that the reliability of cortical representations could have been strongly underestimated.

How do neural networks learn to represent information? Here, we address this question by assuming that neural networks seek to generate an optimal population representation for a fixed linear decoder. We define a loss function for the quality of the population read-out and derive the dynamical equations for both neurons and synapses from the requirement to minimize this loss. The dynamical equations yield a network of integrate-and-fire neurons undergoing Hebbian plasticity. We show that, through learning, initially regular and highly correlated spike trains evolve towards Poisson-distributed and independent spike trains with much lower firing rates. The learning rule drives the network into an asynchronous, balanced regime where all inputs to the network are represented optimally for the given decoder. We show that the network dynamics and synaptic plasticity jointly balance the excitation and inhibition received by each unit as tightly as possible and, in doing so, minimize the prediction error between the inputs and the decoded outputs. In turn, spikes are only signalled whenever this prediction error exceeds a certain value, thereby implementing a predictive coding scheme. Our work suggests that several of the features reported in cortical networks, such as the high trial-to-trial variability, the balance between excitation and inhibition, and spike-timing dependent plasticity, are simply signatures of an efficient, spike-based code.

In many experiments, the data points collected live in high-dimensional observation spaces, yet can be assigned a set of labels or parameters. In electrophysiological recordings, for instance, the responses of populations of neurons generally depend on mixtures of experimentally controlled parameters. The heterogeneity and diversity of these parameter dependencies can make visualization and interpretation of such data extremely difficult. Standard dimensionality reduction techniques such as principal component analysis (PCA) can provide a succinct and complete description of the data, but the description is constructed independent of the relevant task variables and is often hard to interpret. Here, we start with the assumption that a particularly informative description is one that reveals the dependency of the high-dimensional data on the individual parameters. We show how to modify the loss function of PCA so that the principal components seek to capture both the maximum amount of variance about the data, while also depending on a minimum number of parameters. We call this method demixed principal component analysis (dPCA) as the principal components here segregate the parameter dependencies. We phrase the problem as a probabilistic graphical model, and present a fast Expectation-Maximization (EM) algorithm. We demonstrate the use of this algorithm for electrophysiological data and show that it serves to demix the parameter-dependence of a neural population response.

How much information does a neural population convey about a stimulus? Answers to this question are known to strongly depend on the correlation of response variability in neural populations. These noise correlations, however, are essentially immeasurable as the number of parameters in a noise correlation matrix grows quadratically with population size. Here, we suggest to bypass this problem by imposing a parametric model on a noise correlation matrix. Our basic assumption is that noise correlations arise due to common inputs between neurons. On average, noise correlations will therefore reflect signal correlations, which can be measured in neural populations. We suggest an explicit parametric dependency between signal and noise correlations. We show how this dependency can be used to fill the gaps" in noise correlations matrices using an iterative application of the Wishart distribution over positive definitive matrices. We apply our method to data from the primary somatosensory cortex of monkeys performing a two-alternative-forced choice task. We compare the discrimination thresholds read out from the population of recorded neurons with the discrimination threshold of the monkey and show that our method predicts different results than simpler, average schemes of noise correlations."

Neurons that sustain elevated firing in the absence of stimuli have been found in many neural systems. In graded persistent activity, neurons can sustain firing at many levels, suggesting a widely found type of network dynamics in which networks can relax to any one of a continuum of stationary states. The reproduction of these findings in model networks of nonlinear neurons has turned out to be nontrivial. A particularly insightful model has been the "bump attractor," in which a continuous attractor emerges through an underlying symmetry in the network connectivity matrix. This model, however, cannot account for data in which the persistent firing of neurons is a monotonic -- rather than a bell-shaped -- function of a stored variable. Here, we show that the symmetry used in the bump attractor network can be employed to create a whole family of continuous attractor networks, including those with monotonic tuning. Our design is based on tuning the external inputs to networks that have a connectivity matrix with Toeplitz symmetry. In particular, we provide a complete analytical solution of a line attractor network with monotonic tuning and show that for many other networks, the numerical tuning of synaptic weights reduces to the computation of a single parameter.

How do cortical neurons represent the acoustic environment? This ques- tion is often addressed by probing with simple stimuli such as clicks or tone pips. Such stimuli have the advantage of yielding easily interpreted answers, but have the disadvantage that they may fail to uncover complex or higher-order neuronal response properties. Here we adopt an alternative approach, probing neuronal responses with complex acoustic stimuli, including animal vocalizations and music. We have used in vivo whole cell methods in the rat auditory cortex to record subthreshold membrane potential fluctuations elicited by these stimuli. Whole cell recording reveals the total synaptic input to a neuron from all the other neurons in the circuit, instead of just its output—a sparse bi- nary spike train—as in conventional single unit physiological recordings. Whole cell recording thus provides a much richer source of information about the neuron’s response. Many neurons responded robustly and reliably to the complex stimuli in our ensemble. Here we analyze the linear component—the spectro- temporal receptive field (STRF)—of the transformation from the sound (as represented by its time-varying spectrogram) to the neuron’s mem- brane potential. We find that the STRF has a rich dynamical structure, including excitatory regions positioned in general accord with the predic- tion of the simple tuning curve. We also find that in many cases, much of the neuron’s response, although deterministically related to the stimulus, cannot be predicted by the linear component, indicating the presence of as-yet-uncharacterized nonlinear response properties.

We investigate the energy efficiency of signaling mechanisms that transfer information by means of discrete stochastic events, such as the opening or closing of an ion channel. Using a simple model for the generation of graded electrical signals by sodium and potassium channels, we find optimum numbers of channels that maximize energy efficiency. The optima depend on several factors: the relative magnitudes of the signaling cost (current flow through channels), the fixed cost of maintaining the system, the reliability of the input, additional sources of noise, and the relative costs of upstream and downstream mechanisms. We also analyze how the statistics of input signals influence energy efficiency. We find that energy-efficient signal ensembles favor a bimodal distribution of channel activations and contain only a very small fraction of large inputs when energy is scarce. We conclude that when energy use is a significant constraint, trade-offs between information transfer and energy can strongly influence the number of signaling molecules and synapses used by neurons and the manner in which these mechanisms represent information.