Mathematical Modeling seminars

Higher cortical areas carry a wide range of sensory, cognitive, and motor signals supporting complex goal-directed behavior. These signals mix in heterogeneous responses of single neurons, making it difficult to untangle underlying mechanisms. I will present two approaches for revealing interpretable circuit mechanisms from heterogeneous neural responses during cognitive tasks. First, I will show a flexible nonparametric framework for simultaneously inferring population dynamics on single trials and tuning functions of individual neurons to the latent population state. When applied to recordings from the premotor cortex during decision-making, our approach revealed that populations of neurons encoded the same dynamic variable predicting choices, and heterogeneous firing rates resulted from the diverse tuning of single neurons to this decision variable. The inferred dynamics indicated an attractor mechanism for decision computation. Second, I will show an approach for inferring an interpretable network model of a cognitive task—the latent circuit—from neural response data. We developed a theory to causally validate latent circuit mechanisms via patterned perturbations of activity and connectivity in the high-dimensional network. This work opens new possibilities for deriving testable mechanistic hypotheses from complex neural response data.

In this talk, we propose to decipher the activity of neural networks via a “multiply and conquer” approach. This approach considers limit networks made of infinitely many replicas with the same basic neural structure. The key point is that these so-called replica-mean-field networks are in fact simplified, tractable versions of neural networks that retain important features of the finite network structure of interest. The finite size of neuronal populations and synaptic interactions is a core determinant of neural dynamics, being responsible for non-zero correlation in the spiking activity and for finite transition rates between metastable neural states. Theoretically, we develop our replica framework by expanding on ideas from the theory of communication networks rather than from statistical physics to establish Poissonian mean-field limits for spiking networks. Computationally, we leverage our original replica approach to characterize the stationary spiking activity of various network models via reduction to tractable functional equations. We conclude by discussing perspectives about how to use our replica framework to probe nontrivial regimes of spiking correlations and transition rates between metastable neural states.

Activity dependent synaptic plasticity is considered to be a primary mechanism underlying learning and memory. Yet it is unclear whether plasticity rules such as STDP measured in vitro apply in vivo. Network models with STDP predict that activity patterns (e.g., place-cell spatial selectivity) should change much faster than observed experimentally. We address this gap by investigating a nonlinear calcium-based plasticity rule fit to experiments done in physiological conditions. In this model, LTP and LTD result from intracellular calcium transients arising almost exclusively from synchronous coactivation of pre- and postsynaptic neurons. We analytically approximate the full distribution of nonlinear calcium transients as a function of pre- and postsynaptic firing rates, and temporal correlations. This analysis directly relates activity statistics that can be measured in vivo to the changes in synaptic efficacy they cause. Our results highlight that both high-firing rates and temporal correlations can lead to significant changes to synaptic efficacy. Using a mean-field theory, we show that the nonlinear plasticity rule, without any fine-tuning, gives a stable, unimodal synaptic weight distribution characterized by many strong synapses which remain stable over long periods of time, consistent with electrophysiological and behavioral studies. Moreover, our theory explains how memories encoded by strong synapses can be preferentially stabilized by the plasticity rule. We confirmed our analytical results in a spiking recurrent network. Interestingly, although most synapses are weak and undergo rapid turnover, the fraction of strong synapses are sufficient for supporting realistic spiking dynamics and serve to maintain the network’s cluster structure. Our results provide a mechanistic understanding of how stable memories may emerge on the behavioral level from an STDP rule measured in physiological conditions. Furthermore, the plasticity rule we investigate is mathematically equivalent to other learning rules which rely on the statistics of coincidences, so we expect that our formalism will be useful to study other learning processes beyond the calcium-based plasticity rule.

In the past several years, I have been involved in building a biologically realistic model of the monkey visual cortex. Work on one of the input layers (4Ca) of the primary visual cortex (V1) is now nearly complete, and I would like to share some of what I have learned with the community. After a brief overview of the model and its capabilities, I would like to focus on three sets of results that represent three different aspects of the modeling. They are: (i) emergent E-I dynamics in local circuits; (ii) how visual cortical neurons acquire their ability to detect edges and directions of motion, and (iii) a view across the cortical surface: nonequilibrium steady states (in analogy with statistical mechanics) and beyond.

The structural connectivity matrix of synaptic weights between neurons is a critical determinant of overall network function. However, quantitative links between neural network structure and function are complex and subtle. For example, many networks can give rise to similar functional responses, and the same network can function differently depending on context. Whether certain patterns of synaptic connectivity are required to generate specific network-level computations is largely unknown. Here we introduce a geometric framework for identifying synaptic connections required by steady-state responses in recurrent networks of rectified-linear neurons. Assuming that the number of specified response patterns does not exceed the number of input synapses, we analytically calculate all feedforward and recurrent connectivity matrices that can generate the specified responses from the network inputs. We then use this analytical characterization to rigorously analyze the solution space geometry and derive certainty conditions guaranteeing a non-zero synapse between neurons.

In daily life, we consistently make decisions in pursuit of some goal. Many decisions are informed by multiple sources of information. Unfortunately, these sources often provide ambiguous information about what course of action to take. Therefore, determining how the brain integrates information to resolve this ambiguity is key to understanding the neural mechanisms of decision-making. In the domain of time, this topic can be studied by training subjects to predict when a future event will occur based on distinct cues (e.g., tone, light, etc.). If multiple cues are presented simultaneously and their cue-to-event intervals differ (e.g., tone-10s + light-30s), subjects will often expect the event to occur at the average of their intervals. This ‘temporal averaging’ effect is presumably how the timing system resolves ambiguous time-information. The neural mechanisms of temporal averaging are currently unclear. Here, we will propose how temporal averaging could emerge in cortical circuits using a simple modification of a ‘drift-diffusion’ model of timing.

Birdsong emerges when a set of highly interconnected brain areas manage to generate a complex output. The similarities between birdsong production and human speech have positioned songbirds as unique animal models for studying learning and production of this complex motor skill. In this work, we developed a low dimensional model for a neural network in which the variables were the average activities of different neural populations within the nuclei of the song system. This neural network is active during production, perception and learning of birdsong. We performed electrophysiological experiments to record neural activity from one of these nuclei and found that the low dimensional model could reproduce the neural dynamics observed during the experiments. Also, this model could reproduce the respiratory motor patterns used to generate song. We showed that sparse activity in one of the neural nuclei could drive a more complex activity downstream in the neural network. This interdisciplinary work shows how low dimensional neural models can be a valuable tool for studying the emergence of complex motor tasks

Many networks in the nervous system possess an abundance of inhibition, which serves to shape and stabilize neural dynamics. The neurons in such networks exhibit intricate patterns of connectivity whose structure controls the allowed patterns of neural activity. In this work, we examine inhibitory threshold-linear networks whose dynamics are constrained by an underlying directed graph. We develop a set of parameter-independent graph rules that enable us to predict features of the dynamics, such as emergent sequences and dynamic attractors, from properties of the graph. These rules provide a direct link between the structure and function of these networks, and may provide new insights into how connectivity shapes dynamics in real neural circuits.