Characterizing interactions between brain areas is a fundamental goal of systems neuroscience. While such analyses are possible when areas are recorded simultaneously, it is rare to observe all combinations of areas of interest within a single animal or recording session. How can we leverage multi-animal datasets to better understand multi-area interactions? Building on recent progress in large-scale, multi-animal models, we introduce NeuroPaint, a masked autoencoding approach for inferring the dynamics of unobserved brain areas. By training across animals with overlapping subsets of recorded areas, NeuroPaint learns to reconstruct activity in missing areas based on shared structure across individuals. We train and evaluate our approach on both synthetic data and two multi-animal, multi-area Neuropixels datasets. Our results demonstrate that models trained across animals with partial observations can successfully in-paint the dynamics of unrecorded areas, enabling multi-area analyses that transcend the limitations of any single experiment.

When stimulated, neural populations in the visual cortex exhibit fast rhythmic activity with frequencies in the gamma band (30-80 Hz). The gamma rhythm manifests as a broad resonance peak in the power-spectrum of recorded local field potentials, which exhibits various stimulus dependencies. In particular, in macaque primary visual cortex (V1), the gamma peak frequency increases with increasing stimulus contrast. Moreover, this contrast dependence is local: when contrast varies smoothly over visual space, the gamma peak frequency in each cortical column is controlled by the local contrast in that column's receptive field. No parsimonious mechanistic explanation for these contrast dependencies of V1 gamma oscillations has been proposed. The stabilized supralinear network (SSN) is a mechanistic model of cortical circuits that has accounted for a range of visual cortical response nonlinearities and contextual modulations, as well as their contrast dependence. Here, we begin by showing that a reduced SSN model without retinotopy robustly captures the contrast dependence of gamma peak frequency, and provides a mechanistic explanation for this effect based on the observed non-saturating and supralinear input-output function of V1 neurons. Given this result, the local dependence on contrast can trivially be captured in a retinotopic SSN which however lacks horizontal synaptic connections between its cortical columns. However, long-range horizontal connections in V1 are in fact strong, and underlie contextual modulation effects such as surround suppression. We thus explored whether a retinotopically organized SSN model of V1 with strong excitatory horizontal connections can exhibit both surround suppression and the local contrast dependence of gamma peak frequency. We found that retinotopic SSNs can account for both effects, but only when the horizontal excitatory projections are composed of two components with different patterns of spatial fall-off with distance: a short-range component that only targets the source column, combined with a long-range component that targets columns neighboring the source column. We thus make a specific qualitative prediction for the spatial structure of horizontal connections in macaque V1, consistent with the columnar structure of cortex.

We study a rate-model neural network composed of excitatory and inhibitory neurons in which neuronal input-output functions are power laws with a power greater than 1, as observed in primary visual cortex. This supralinear input-output function leads to supralinear summation of network responses to multiple inputs for weak inputs. We show that for stronger inputs, which would drive the excitatory subnetwork to instability, the network will dynamically stabilize provided feedback inhibition is sufficiently strong. For a wide range of network and stimulus parameters, this dynamic stabilization yields a transition from supralinear to sublinear summation of network responses to multiple inputs. We compare this to the dynamic stabilization in the balanced network, which yields only linear behavior. We more exhaustively analyze the two-dimensional case of one excitatory and one inhibitory population. We show that in this case, dynamic stabilization will occur whenever the determinant of the weight matrix is positive and the inhibitory time constant is sufficiently small, and analyze the conditions for supersaturation, or decrease of firing rates with increasing stimulus contrast (which represents increasing input firing rates). In work to be presented elsewhere, we have found that this transition from supralinear to sublinear summation can explain a wide variety of nonlinearities in cerebral cortical processing.

We demonstrate the mathematical equivalence of two commonly used forms of firing rate model equations for neural networks. In addition, we show that what is commonly interpreted as the firing rate in one form of model may be better interpreted as a low-pass-filtered firing rate, and we point out a conductance-based firing rate model.

Loopy belief propagation performs approximate inference on graphical models with loops. One might hope to compensate for the approximation by adjusting model parameters. Learning algorithms for this purpose have been explored previously, and the claim has been made that every set of locally consistent marginals can arise from belief propagation run on a graphical model. On the contrary, here we show that many probability distributions have marginals that cannot be reached by belief propagation using any set of model parameters or any learning algorithm. We call such marginals `unbelievable.' This problem occurs whenever the Hessian of the Bethe free energy is not positive-definite at the target marginals. All learning algorithms for belief propagation necessarily fail in these cases, producing beliefs or sets of beliefs that may even be worse than the pre-learning approximation. We then show that averaging inaccurate beliefs, each obtained from belief propagation using model parameters perturbed about some learned mean values, can achieve the unbelievable marginals.

Signal-to-noise ratios in physical systems can be significantly degraded if the outputs of the systems are highly variable. Biological processes for which highly stereotyped signal generations are necessary features appear to have reduced their signal variabilities by employing multiple processing steps. To better understand why this multistep cascade structure might be desirable, we prove that the reliability of a signal generated by a multistate system with no memory (i.e., a Markov chain) is maximal if and only if the system topology is such that the process steps irreversibly through each state, with transition rates chosen such that an equal fraction of the total signal is generated in each state. Furthermore, our result indicates that by increasing the number of states, it is possible to arbitrarily increase the reliability of the system. In a physical system, however, an energy cost is associated with maintaining irreversible transitions, and this cost increases with the number of such transitions (i.e., the number of states). Thus, an infinite-length chain, which would be perfectly reliable, is infeasible. To model the effects of energy demands on the maximally reliable solution, we numerically optimize the topology under two distinct energy functions that penalize either irreversible transitions or incommunicability between states, respectively. In both cases, the solutions are essentially irreversible linear chains, but with upper bounds on the number of states set by the amount of available energy. We therefore conclude that a physical system for which signal reliability is important should employ a linear architecture, with the number of states (and thus the reliability) determined by the intrinsic energy constraints of the system.

Cortical synaptic plasticity depends on the relative timing of pre- and postsynaptic spikes and also on the temporal pattern of presynaptic spikes and of postsynaptic spikes. We study the hypothesis that cortical synap- tic plasticity does not associate individual spikes, but rather whole fir- ing episodes, and depends only on when these episodes start and how long they last, but as little as possible on the timing of individual spikes. Here we present the mathematical background for such a study. Stan- dard methods from hidden Markov models are used to define what “fir- ing episodes” are. Estimating the probability of being in such an episode requires not only the knowledge of past spikes, but also of future spikes. We show how to construct a causal learning rule, which depends only on past spikes, but associates pre- and postsynaptic firing episodes as if it also knew future spikes. We also show that this learning rule agrees with some features of synaptic plasticity in superficial layers of rat visual cortex (Froemke and Dan, Nature 416:433, 2002).

A simple model of correlation-based synaptic plasticity via axonal sprouting and retraction (Elliott, Howarth, & Shadbolt, 1996a) is shown to be equivalent to the class of correlation-based models (Miller, Keller, & Stryker, 1989), although these were formulated in terms of weight modification of anatomically fixed synapses. Both models maximize the same measure of synaptic correlation, subject to certain constraints on connectivity. Thus, the analyses of the correlation-based models suffice to characterize the behavior of the sprouting-and-retraction model. More detailed models are needed for theoretical distinctions to be drawn between plasticity via sprouting and retraction, weight modification, or a combination. The model of Elliott et al. involves stochastic search through allowed weight patterns for those that improve correlations. That of Miller et al. instead follows dynamical equations that determine continuous changes of the weights that improve correlations. The identity of these two approaches is shown to depend on the use of subtractive constraint enforcement in the models of Miller et al. More generally, to model the idea that neural development acts to maximize some measure of correlation subject to a constraint on the summed synaptic weight, the constraint must be enforced subtractively in a dynamical model.

To understand the interspike interval (ISI) variability displayed by visual cortical neurons (Softky & Koch, 1993), it is critical to examine the dynamics of their neuronal integration, as well as the variability in their synaptic input current. Most previous models have focused on the latter factor. We match a simple integrate-and-fire model to the experimentally measured integrative properties of cortical regular spiking cells (McCormick, Connors, Lighthall, & Prince, 1985). After setting RC parameters, the post-spike voltage reset is set to match experimental measurements of neuronal gain (obtained from in vitro plots of firing frequency versus injected current). Examination of the resulting model leads to an intuitive picture of neuronal integration that unifies the seemingly contradictory 1/square root of N and random walk pictures that have previously been proposed. When ISIs are dominated by postspike recovery, 1/square root of N arguments hold and spiking is regular; after the "memory" of the last spike becomes negligible, spike threshold crossing is caused by input variance around a steady state and spiking is Poisson. In integrate-and-fire neurons matched to cortical cell physiology, steady-state behavior is predominant, and ISIs are highly variable at all physiological firing rates and for a wide range of inhibitory and excitatory inputs.

Models of unsupervised, correlation-based (Hebbian) synaptic plasticity are typically unstable: either all synapses grow until each reaches the maximum allowed strength, or all synapses decay to zero strength. A common method of avoiding these outcomes is to use a constraint that conserves or limits the total synaptic strength over a cell. We study the dynamic effects of such constraints. Two methods of enforcing a constraint are distinguished, multiplicative and subtractive. For otherwise linear learning rules, multiplicative enforcement of a constraint results in dynamics that converge to the principal eigenvector of the operator determining unconstrained synaptic development. Subtractive enforcement, in contrast, typically leads to a final state in which almost all synaptic strengths reach either the maximum or minimum allowed value. This final state is often dominated by weight configurations other than the principal eigenvector of the unconstrained operator. Multiplicative enforcement yields a “graded” receptive field in which most mutually correlated inputs are represented, whereas subtractive enforcement yields a receptive field that is “sharpened” to a subset of maximally correlated inputs. If two equivalent input populations (e.g., two eyes) innervate a common target, multiplicative enforcement prevents their segregation (ocular dominance segregation) when the two populations are weakly correlated; whereas subtractive enforcement allows segregation under these circumstances. These results may be used to understand constraints both over output cells and over input cells. A variety of rules that can implement constrained dynamics are discussed.

Linsker has reported the development of center-surround receptive fields and oriented receptive fields in simulations of a Hebb-type equation in a linear network. The dynamics of the learning rule are analyzed in terms of the eigenvectors of the covariance matrix of cell activities. Analytic and computational results for Linsker's covariance matrices, and some general theorems, lead to an explanation of the emergence of center-surround and certain oriented structures. We estimate criteria for the parameter regime in which center-surround structures emerge.

A linear Hebbian equation for synaptic plasticity is derived from a more complex, nonlinear model by considering the initial development of the difference between two equivalent excitatory projections. This provides a justification for the use of such a simple equation to model activity-dependent neural development and plasticity, and allows analysis of the biological origins of the terms in the equation. Connections to previously published models are discussed.