Rapid memory encoding in a recurrent network model with behavioral time scale synaptic plasticity.

Episodic memories are formed after a single exposure to novel stimuli. The plasticity mechanisms underlying such fast learning still remain largely unknown. Recently, it was shown that cells in area CA1 of the hippocampus of mice could form or shift their place fields after a single traversal of a virtual linear track. In-vivo intracellular recordings in CA1 cells revealed that previously silent inputs from CA3 could be switched on when they occurred within a few seconds of a dendritic plateau potential (PP) in the post-synaptic cell, a phenomenon dubbed Behavioral Time-scale Plasticity (BTSP). A recently developed computational framework for BTSP in which the dynamics of synaptic traces related to the pre-synaptic activity and post-synaptic PP are explicitly modelled, can account for experimental findings. Here we show that this model of plasticity can be further simplified to a 1D map which describes changes to the synaptic weights after a single trial. We use a temporally symmetric version of this map to study the storage of a large number of spatial memories in a recurrent network, such as CA3. Specifically, the simplicity of the map allows us to calculate the correlation of the synaptic weight matrix with any given past environment analytically. We show that the calculated memory trace can be used to predict the emergence and stability of bump attractors in a high dimensional neural network model endowed with BTSP.

Flexible integration of continuous sensory evidence in perceptual estimation tasks.

Temporal accumulation of evidence is crucial for making accurate judgments based on noisy or ambiguous sensory input. The integration process leading to categorical decisions is thought to rely on competition between neural populations, each encoding a discrete categorical choice. How recurrent neural circuits integrate evidence for continuous perceptual judgments is unknown. Here, we show that a continuous bump attractor network can integrate a circular feature, such as stimulus direction, nearly optimally. As required by optimal integration, the population activity of the network unfolds on a two-dimensional manifold, in which the position of the network's activity bump tracks the stimulus average, and, simultaneously, the bump amplitude tracks stimulus uncertainty. Moreover, the temporal weighting of sensory evidence by the network depends on the relative strength of the stimulus compared to the internally generated bump dynamics, yielding either early (primacy), uniform, or late (recency) weighting. The model can flexibly switch between these regimes by changing a single control parameter, the global excitatory drive. We show that this mechanism can quantitatively explain individual temporal weighting profiles of human observers, and we validate the model prediction that temporal weighting impacts reaction times. Our findings point to continuous attractor dynamics as a plausible neural mechanism underlying stimulus integration in perceptual estimation tasks.

Flexible categorization in perceptual decision making.

Perceptual decisions rely on accumulating sensory evidence. This computation has been studied using either drift diffusion models or neurobiological network models exhibiting winner-take-all attractor dynamics. Although both models can account for a large amount of data, it remains unclear whether their dynamics are qualitatively equivalent. Here we show that in the attractor model, but not in the drift diffusion model, an increase in the stimulus fluctuations or the stimulus duration promotes transitions between decision states. The increase in the number of transitions leads to a crossover between weighting mostly early evidence (primacy) to weighting late evidence (recency), a prediction we validate with psychophysical data. Between these two limiting cases, we found a novel flexible categorization regime, in which fluctuations can reverse initially-incorrect categorizations. This reversal asymmetry results in a non-monotonic psychometric curve, a distinctive feature of the attractor model. Our findings point to correcting decision reversals as an important feature of perceptual decision making.

Exact firing rate model reveals the differential effects of chemical versus electrical synapses in spiking networks.

Chemical and electrical synapses shape the dynamics of neuronal networks. Numerous theoretical studies have investigated how each of these types of synapses contributes to the generation of neuronal oscillations, but their combined effect is less understood. This limitation is further magnified by the impossibility of traditional neuronal mean-field models-also known as firing rate models or firing rate equations-to account for electrical synapses. Here, we introduce a firing rate model that exactly describes the mean-field dynamics of heterogeneous populations of quadratic integrate-and-fire (QIF) neurons with both chemical and electrical synapses. The mathematical analysis of the firing rate model reveals a well-established bifurcation scenario for networks with chemical synapses, characterized by a codimension-2 cusp point and persistent states for strong recurrent excitatory coupling. The inclusion of electrical coupling generally implies neuronal synchrony by virtue of a supercritical Hopf bifurcation. This transforms the cusp scenario into a bifurcation scenario characterized by three codimension-2 points (cusp, Takens-Bogdanov, and saddle-node separatrix loop), which greatly reduces the possibility for persistent states. This is generic for heterogeneous QIF networks with both chemical and electrical couplings. Our results agree with several numerical studies on the dynamics of large networks of heterogeneous spiking neurons with electrical and chemical couplings.

Firing rate distributions in spiking networks with heterogeneous connectivity.

Mean-field theory for networks of spiking neurons based on the so-called diffusion approximation has been used to calculate certain measures of neuronal activity which can be compared with experimental data. This includes the distribution of firing rates across the network. However, the theory in its current form applies only to networks in which there is relatively little heterogeneity in the number of incoming and outgoing connections per neuron. Here we extend this theory to include networks with arbitrary degree distributions. Furthermore, the theory takes into account correlations in the in-degree and out-degree of neurons, which would arise, e.g., in the case of networks with hublike neurons. Finally, we show that networks with broad and positively correlated degrees can generate a large-amplitude sustained response to transient stimuli which does not occur in more homogeneous networks.

Drift-diffusion models for multiple-alternative forced-choice decision making.

The canonical computational model for the cognitive process underlying two-alternative forced-choice decision making is the so-called drift-diffusion model (DDM). In this model, a decision variable keeps track of the integrated difference in sensory evidence for two competing alternatives. Here I extend the notion of a drift-diffusion process to multiple alternatives. The competition between n alternatives takes place in a linear subspace of [Formula: see text] dimensions; that is, there are [Formula: see text] decision variables, which are coupled through correlated noise sources. I derive the multiple-alternative DDM starting from a system of coupled, linear firing rate equations. I also show that a Bayesian sequential probability ratio test for multiple alternatives is, in fact, equivalent to these same linear DDMs, but with time-varying thresholds. If the original neuronal system is nonlinear, one can once again derive a model describing a lower-dimensional diffusion process. The dynamics of the nonlinear DDM can be recast as the motion of a particle on a potential, the general form of which is given analytically for an arbitrary number of alternatives.

Theta-modulation drives the emergence of connectivity patterns underlying replay in a network model of place cells.

Place cells of the rodent hippocampus fire action potentials when the animal traverses a particular spatial location in any environment. Therefore for any given trajectory one observes a repeatable sequence of place cell activations. When the animal is quiescent or sleeping, one can observe similar sequences of activation known as replay, which underlie the process of memory consolidation. However, it remains unclear how replay is generated. Here we show how a temporally asymmetric plasticity rule during spatial exploration gives rise to spontaneous replay in a model network by shaping the recurrent connectivity to reflect the topology of the learned environment. Crucially, the rate of this encoding is strongly modulated by ongoing rhythms. Oscillations in the theta range optimize learning by generating repeated pre-post pairings on a time-scale commensurate with the window for plasticity, while lower and higher frequencies generate learning rates which are lower by orders of magnitude.

Network mechanisms underlying the role of oscillations in cognitive tasks.

Oscillatory activity robustly correlates with task demands during many cognitive tasks. However, not only are the network mechanisms underlying the generation of these rhythms poorly understood, but it is also still unknown to what extent they may play a functional role, as opposed to being a mere epiphenomenon. Here we study the mechanisms underlying the influence of oscillatory drive on network dynamics related to cognitive processing in simple working memory (WM), and memory recall tasks. Specifically, we investigate how the frequency of oscillatory input interacts with the intrinsic dynamics in networks of recurrently coupled spiking neurons to cause changes of state: the neuronal correlates of the corresponding cognitive process. We find that slow oscillations, in the delta and theta band, are effective in activating network states associated with memory recall. On the other hand, faster oscillations, in the beta range, can serve to clear memory states by resonantly driving transient bouts of spike synchrony which destabilize the activity. We leverage a recently derived set of exact mean-field equations for networks of quadratic integrate-and-fire neurons to systematically study the bifurcation structure in the periodically forced spiking network. Interestingly, we find that the oscillatory signals which are most effective in allowing flexible switching between network states are not smooth, pure sinusoids, but rather burst-like, with a sharp onset. We show that such periodic bursts themselves readily arise spontaneously in networks of excitatory and inhibitory neurons, and that the burst frequency can be tuned via changes in tonic drive. Finally, we show that oscillations in the gamma range can actually stabilize WM states which otherwise would not persist.

Firing rate equations require a spike synchrony mechanism to correctly describe fast oscillations in inhibitory networks.

Recurrently coupled networks of inhibitory neurons robustly generate oscillations in the gamma band. Nonetheless, the corresponding Wilson-Cowan type firing rate equation for such an inhibitory population does not generate such oscillations without an explicit time delay. We show that this discrepancy is due to a voltage-dependent spike-synchronization mechanism inherent in networks of spiking neurons which is not captured by standard firing rate equations. Here we investigate an exact low-dimensional description for a network of heterogeneous canonical Class 1 inhibitory neurons which includes the sub-threshold dynamics crucial for generating synchronous states. In the limit of slow synaptic kinetics the spike-synchrony mechanism is suppressed and the standard Wilson-Cowan equations are formally recovered as long as external inputs are also slow. However, even in this limit synchronous spiking can be elicited by inputs which fluctuate on a time-scale of the membrane time-constant of the neurons. Our meanfield equations therefore represent an extension of the standard Wilson-Cowan equations in which spike synchrony is also correctly described.

UP-DOWN cortical dynamics reflect state transitions in a bistable network.

In the idling brain, neuronal circuits transition between periods of sustained firing (UP state) and quiescence (DOWN state), a pattern the mechanisms of which remain unclear. Here we analyzed spontaneous cortical population activity from anesthetized rats and found that UP and DOWN durations were highly variable and that population rates showed no significant decay during UP periods. We built a network rate model with excitatory (E) and inhibitory (I) populations exhibiting a novel bistable regime between a quiescent and an inhibition-stabilized state of arbitrarily low rate. Fluctuations triggered state transitions, while adaptation in E cells paradoxically caused a marginal decay of E-rate but a marked decay of I-rate in UP periods, a prediction that we validated experimentally. A spiking network implementation further predicted that DOWN-to-UP transitions must be caused by synchronous high-amplitude events. Our findings provide evidence of bistable cortical networks that exhibit non-rhythmic state transitions when the brain rests.

On the Structure of Cortical Microcircuits Inferred from Small Sample Sizes.

The structure in cortical microcircuits deviates from what would be expected in a purely random network, which has been seen as evidence of clustering. To address this issue, we sought to reproduce the nonrandom features of cortical circuits by considering several distinct classes of network topology, including clustered networks, networks with distance-dependent connectivity, and those with broad degree distributions. To our surprise, we found that all of these qualitatively distinct topologies could account equally well for all reported nonrandom features despite being easily distinguishable from one another at the network level. This apparent paradox was a consequence of estimating network properties given only small sample sizes. In other words, networks that differ markedly in their global structure can look quite similar locally. This makes inferring network structure from small sample sizes, a necessity given the technical difficulty inherent in simultaneous intracellular recordings, problematic. We found that a network statistic called the sample degree correlation (SDC) overcomes this difficulty. The SDC depends only on parameters that can be estimated reliably given small sample sizes and is an accurate fingerprint of every topological family. We applied the SDC criterion to data from rat visual and somatosensory cortex and discovered that the connectivity was not consistent with any of these main topological classes. However, we were able to fit the experimental data with a more general network class, of which all previous topologies were special cases. The resulting network topology could be interpreted as a combination of physical spatial dependence and nonspatial, hierarchical clustering.SIGNIFICANCE STATEMENT The connectivity of cortical microcircuits exhibits features that are inconsistent with a simple random network. Here, we show that several classes of network models can account for this nonrandom structure despite qualitative differences in their global properties. This apparent paradox is a consequence of the small numbers of simultaneously recorded neurons in experiment: when inferred via small sample sizes, many networks may be indistinguishable despite being globally distinct. We develop a connectivity measure that successfully classifies networks even when estimated locally with a few neurons at a time. We show that data from rat cortex is consistent with a network in which the likelihood of a connection between neurons depends on spatial distance and on nonspatial, asymmetric clustering.

Mean-field equations for neuronal networks with arbitrary degree distributions.

The emergent dynamics in networks of recurrently coupled spiking neurons depends on the interplay between single-cell dynamics and network topology. Most theoretical studies on network dynamics have assumed simple topologies, such as connections that are made randomly and independently with a fixed probability (Erdös-Rényi network) (ER) or all-to-all connected networks. However, recent findings from slice experiments suggest that the actual patterns of connectivity between cortical neurons are more structured than in the ER random network. Here we explore how introducing additional higher-order statistical structure into the connectivity can affect the dynamics in neuronal networks. Specifically, we consider networks in which the number of presynaptic and postsynaptic contacts for each neuron, the degrees, are drawn from a joint degree distribution. We derive mean-field equations for a single population of homogeneous neurons and for a network of excitatory and inhibitory neurons, where the neurons can have arbitrary degree distributions. Through analysis of the mean-field equations and simulation of networks of integrate-and-fire neurons, we show that such networks have potentially much richer dynamics than an equivalent ER network. Finally, we relate the degree distributions to so-called cortical motifs.

Synchrony-induced modes of oscillation of a neural field model.

We investigate the modes of oscillation of heterogeneous ring networks of quadratic integrate-and-fire (QIF) neurons with nonlocal, space-dependent coupling. Perturbations of the equilibrium state with a particular wave number produce transient standing waves with a specific temporal frequency, analogously to those in a tense string. In the neuronal network, the equilibrium corresponds to a spatially homogeneous, asynchronous state. Perturbations of this state excite the network's oscillatory modes, which reflect the interplay of episodes of synchronous spiking with the excitatory-inhibitory spatial interactions. In the thermodynamic limit, an exact low-dimensional neural field model describing the macroscopic dynamics of the network is derived. This allows us to obtain formulas for the Turing eigenvalues of the spatially homogeneous state and hence to obtain its stability boundary. We find that the frequency of each Turing mode depends on the corresponding Fourier coefficient of the synaptic pattern of connectivity. The decay rate instead is identical for all oscillation modes as a consequence of the heterogeneity-induced desynchronization of the neurons. Finally, we numerically compute the spectrum of spatially inhomogeneous solutions branching from the Turing bifurcation, showing that similar oscillatory modes operate in neural bump states and are maintained away from onset.

Oscillations in the bistable regime of neuronal networks.

Bistability between attracting fixed points in neuronal networks has been hypothesized to underlie persistent activity observed in several cortical areas during working memory tasks. In network models this kind of bistability arises due to strong recurrent excitation, sufficient to generate a state of high activity created in a saddle-node (SN) bifurcation. On the other hand, canonical network models of excitatory and inhibitory neurons (E-I networks) robustly produce oscillatory states via a Hopf (H) bifurcation due to the E-I loop. This mechanism for generating oscillations has been invoked to explain the emergence of brain rhythms in the ß to γ bands. Although both bistability and oscillatory activity have been intensively studied in network models, there has not been much focus on the coincidence of the two. Here we show that when oscillations emerge in E-I networks in the bistable regime, their phenomenology can be explained to a large extent by considering coincident SN and H bifurcations, known as a codimension two Takens-Bogdanov bifurcation. In particular, we find that such oscillations are not composed of a stable limit cycle, but rather are due to noise-driven oscillatory fluctuations. Furthermore, oscillations in the bistable regime can, in principle, have arbitrarily low frequency.

Sensory integration dynamics in a hierarchical network explains choice probabilities in cortical area MT.

Neuronal variability in sensory cortex predicts perceptual decisions. This relationship, termed choice probability (CP), can arise from sensory variability biasing behaviour and from top-down signals reflecting behaviour. To investigate the interaction of these mechanisms during the decision-making process, we use a hierarchical network model composed of reciprocally connected sensory and integration circuits. Consistent with monkey behaviour in a fixed-duration motion discrimination task, the model integrates sensory evidence transiently, giving rise to a decaying bottom-up CP component. However, the dynamics of the hierarchical loop recruits a concurrently rising top-down component, resulting in sustained CP. We compute the CP time-course of neurons in the medial temporal area (MT) and find an early transient component and a separate late contribution reflecting decision build-up. The stability of individual CPs and the dynamics of noise correlations further support this decomposition. Our model provides a unified understanding of the circuit dynamics linking neural and behavioural variability.

Efficient partitioning of memory systems and its importance for memory consolidation.

Long-term memories are likely stored in the synaptic weights of neuronal networks in the brain. The storage capacity of such networks depends on the degree of plasticity of their synapses. Highly plastic synapses allow for strong memories, but these are quickly overwritten. On the other hand, less labile synapses result in long-lasting but weak memories. Here we show that the trade-off between memory strength and memory lifetime can be overcome by partitioning the memory system into multiple regions characterized by different levels of synaptic plasticity and transferring memory information from the more to less plastic region. The improvement in memory lifetime is proportional to the number of memory regions, and the initial memory strength can be orders of magnitude larger than in a non-partitioned memory system. This model provides a fundamental computational reason for memory consolidation processes at the systems level.

On the distribution of firing rates in networks of cortical neurons.

The distribution of in vivo average firing rates within local cortical networks has been reported to be highly skewed and long tailed. The distribution of average single-cell inputs, conversely, is expected to be Gaussian by the central limit theorem. This raises the issue of how a skewed distribution of firing rates might result from a symmetric distribution of inputs. We argue that skewed rate distributions are a signature of the nonlinearity of the in vivo f-I curve. During in vivo conditions, ongoing synaptic activity produces significant fluctuations in the membrane potential of neurons, resulting in an expansive nonlinearity of the f-I curve for low and moderate inputs. Here, we investigate the effects of single-cell and network parameters on the shape of the f-I curve and, by extension, on the distribution of firing rates in randomly connected networks.

The role of degree distribution in shaping the dynamics in networks of sparsely connected spiking neurons.

Neuronal network models often assume a fixed probability of connection between neurons. This assumption leads to random networks with binomial in-degree and out-degree distributions which are relatively narrow. Here I study the effect of broad degree distributions on network dynamics by interpolating between a binomial and a truncated power-law distribution for the in-degree and out-degree independently. This is done both for an inhibitory network (I network) as well as for the recurrent excitatory connections in a network of excitatory and inhibitory neurons (EI network). In both cases increasing the width of the in-degree distribution affects the global state of the network by driving transitions between asynchronous behavior and oscillations. This effect is reproduced in a simplified rate model which includes the heterogeneity in neuronal input due to the in-degree of cells. On the other hand, broadening the out-degree distribution is shown to increase the fraction of common inputs to pairs of neurons. This leads to increases in the amplitude of the cross-correlation (CC) of synaptic currents. In the case of the I network, despite strong oscillatory CCs in the currents, CCs of the membrane potential are low due to filtering and reset effects, leading to very weak CCs of the spike-count. In the asynchronous regime of the EI network, broadening the out-degree increases the amplitude of CCs in the recurrent excitatory currents, while CC of the total current is essentially unaffected as are pairwise spiking correlations. This is due to a dynamic balance between excitatory and inhibitory synaptic currents. In the oscillatory regime, changes in the out-degree can have a large effect on spiking correlations and even on the qualitative dynamical state of the network.

The statistics of repeating patterns of cortical activity can be reproduced by a model network of stochastic binary neurons.

Calcium imaging of the spontaneous activity in cortical slices has revealed repeating spatiotemporal patterns of transitions between so-called down states and up states (Ikegaya et al., 2004). Here we fit a model network of stochastic binary neurons to data from these experiments, and in doing so reproduce the distributions of such patterns. We use two versions of this model: (1) an unconnected network in which neurons are activated as independent Poisson processes; and (2) a network with an interaction matrix, estimated from the data, representing effective interactions between the neurons. The unconnected model (model 1) is sufficient to account for the statistics of repeating patterns in 11 of the 15 datasets studied. Model 2, with interactions between neurons, is required to account for pattern statistics of the remaining four. Three of these four datasets are the ones that contain the largest number of transitions, suggesting that long datasets are in general necessary to render interactions statistically visible. We then study the topology of the matrix of interactions estimated for these four datasets. For three of the four datasets, we find sparse matrices with long-tailed degree distributions and an overrepresentation of certain network motifs. The remaining dataset exhibits a strongly interconnected, spatially localized subgroup of neurons. In all cases, we find that interactions between neurons facilitate the generation of long patterns that do not repeat exactly.

Neurobiological models of two-choice decision making can be reduced to a one-dimensional nonlinear diffusion equation.

The response behaviors in many two-alternative choice tasks are well described by so-called sequential sampling models. In these models, the evidence for each one of the two alternatives accumulates over time until it reaches a threshold, at which point a response is made. At the neurophysiological level, single neuron data recorded while monkeys are engaged in two-alternative choice tasks are well described by winner-take-all network models in which the two choices are represented in the firing rates of separate populations of neurons. Here, we show that such nonlinear network models can generally be reduced to a one-dimensional nonlinear diffusion equation, which bears functional resemblance to standard sequential sampling models of behavior. This reduction gives the functional dependence of performance and reaction-times on external inputs in the original system, irrespective of the system details. What is more, the nonlinear diffusion equation can provide excellent fits to behavioral data from two-choice decision making tasks by varying these external inputs. This suggests that changes in behavior under various experimental conditions, e.g. changes in stimulus coherence or response deadline, are driven by internal modulation of afferent inputs to putative decision making circuits in the brain. For certain model systems one can analytically derive the nonlinear diffusion equation, thereby mapping the original system parameters onto the diffusion equation coefficients. Here, we illustrate this with three model systems including coupled rate equations and a network of spiking neurons.