Mathematical Modeling seminars

Brain networks exist within the confines of resource limitations. As a result, a brain network must overcome metabolic costs of growing and sustaining the network within its physical space, while simultaneously implementing its required information processing. To observe the effect of these processes, we introduce the spatially-embedded recurrent neural network (seRNN). seRNNs learn basic task-related inferences while existing within a 3D Euclidean space, where the communication of constituent neurons is constrained by a sparse connectome. We find that seRNNs, similar to primate cerebral cortices, naturally converge on solving inferences using modular small-world networks, in which functionally similar units spatially configure themselves to utilize an energetically-efficient mixed-selective code. As all these features emerge in unison, seRNNs reveal how many common structural and functional brain motifs are strongly intertwined and can be attributed to basic biological optimization processes. seRNNs can serve as model systems to bridge between structural and functional research communities to move neuroscientific understanding forward.

Vector-space models are used frequently to compare similarity and dimensionality among entity concepts. What happens when we apply these models to relational concepts? What is the evidence that such models do apply to relational concepts? If we use such a model, then one implication is that maximizing surface feature variation should improve relational concept learning. For example, in STEM instruction, the effectiveness of teaching by analogy is often limited by students’ focus on superficial features of the source and target exemplars. However, in contrast to the prediction of the vector-space computational model, the strategy of progressive alignment (moving from perceptually similar to different targets) has been suggested to address this issue (Gentner & Hoyos, 2017), and human behavioral evidence has shown benefits from progressive alignment. Here I will present some preliminary data that supports the computational approach. Participants were explicitly instructed to match stimuli based on relations while perceptual similarity of stimuli varied parametrically. We found that lower perceptual similarity reduced accurate relational matching. This finding demonstrates that perceptual similarity may interfere with relational judgements, but also hints at why progressive alignment maybe effective. These are preliminary, exploratory data and I to hope receive feedback on the framework and to start a discussion in a group on the utility of vector-space models for relational concepts in general.

A signature feature of cortical circuits is the irregularity of neuronal firing, which manifests itself in the high temporal variability of spiking and the broad distribution of rates. Theoretical works have shown that this feature emerges dynamically in network models if coupling between cells is strong, i.e. if the mean number of synapses per neuron K is large and synaptic efficacy is of order 1/\sqrt{K}. However, the degree to which these models capture the mechanisms underlying neuronal firing in cortical circuits is not fully understood. Results have been derived using neuron models with current-based synapses, i.e. neglecting the dependence of synaptic current on the membrane potential, and an understanding of how irregular firing emerges in models with conductance-based synapses is still lacking. Moreover, at odds with the nonlinear responses to multiple stimuli observed in cortex, network models with strongly coupled cells respond linearly to inputs. In this talk, I will discuss the emergence of irregular firing and nonlinear response in networks of leaky integrate-and-fire neurons. First, I will show that, when synapses are conductance-based, irregular firing emerges if synaptic efficacy is of order 1/\log(K) and, unlike in current-based models, persists even under the large heterogeneity of connections which has been reported experimentally. I will then describe an analysis of neural responses as a function of coupling strength and show that, while a linear input-output relation is ubiquitous at strong coupling, nonlinear responses are prominent at moderate coupling. I will conclude by discussing experimental evidence of moderate coupling and loose balance in the mouse cortex.

Decades of research have investigated the relationship between perception of time and heart rate with often mixed results. In search of such a relationship, I will present my far journey between two projects: from time perception in the realistic VR experience of crowded subway trips in the order of minutes (project 1); to the perceived duration of sub-second white noise tones (project 2). Heart rate had multiple concurrent relationships with subjective temporal distortions for the sub-second tones, while the effects were lacking or weak for the supra-minute subway trips. What does the heart have to do with sub-second time perception? We addressed this question with a cardiac drift-diffusion model, demonstrating the sensory accumulation of temporal evidence as a function of heart rate.

One of the fundamental functions of the brain is to flexibly plan and control movement production at different timescales in order to efficiently shape structured behaviors. I will present research investigating how these complex computations are performed in the mammalian brain, with an emphasis on autonomous motor control. Specifically, I will focus on the mechanisms supporting efficient interfacing between 'higher-level' planning commands and 'lower-level' motor cortical dynamics that ultimately drive muscles. I will take advantage of the fact that the anatomy of the circuits underlying motor control is well known. It notably involves the primary motor cortex, a recurrent network that generates learned commands to drive muscles while interacting through loops with thalamic neurons that lack recurrent excitation. Using an analytically tractable model that incorporates these architectural constraints, I will explain how this motor circuit can implement a form of efficient modularity by combining (i) plastic thalamocortical loops that are movement-specific and (ii) shared hardwired circuits. I will show that this modular architecture can balance two different objectives: first, supporting the flexible recombination of an extensible library of re-usable motor primitives; and second, promoting the efficient use of neural resources by taking advantage of shared connections between modules. I will finally show that these insights are relevant for designing artificial neural networks able to flexibly and robustly compose hierarchical analog behaviors from a library of motor primitives. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2023-01-11. Recording duration: 00:49:52.

An influential theory in systems neuroscience suggests that brain function can be understood through low-dimensional dynamics [Vyas et al 2020]. However, a challenge in this framework is that a single computational task may involve a range of dynamic processes. To understand which processes are at play in the brain, it is important to use data on neural activity to constrain models. In this study, we present a method for extracting low-dimensional dynamics from data using low-rank recurrent neural networks (lrRNNs), a highly expressive and understandable type of model [Mastrogiuseppe & Ostojic 2018, Dubreuil, Valente et al. 2022]. We first test our approach using synthetic data created from full-rank RNNs that have been trained on various brain tasks. We find that lrRNNs fitted to neural activity allow us to identify the collective computational processes and make new predictions for inactivations in the original RNNs. We then apply our method to data recorded from the prefrontal cortex of primates during a context-dependent decision-making task. Our approach enables us to assign computational roles to the different latent variables and provides a mechanistic model of the recorded dynamics, which can be used to perform in silico experiments like inactivations and provide testable predictions.

Understanding Human ability to learn novel concepts from just a few sensory experiences is a fundamental problem in cognitive neuroscience. I will describe a recent work with Ben Sorcher and Surya Ganguli (PNAS, October 2022) in which we propose a simple, biologically plausible, and mathematically tractable neural mechanism for few-shot learning of naturalistic concepts. We posit that the concepts that can be learned from few examples are defined by tightly circumscribed manifolds in the neural firing-rate space of higher-order sensory areas. Discrimination between novel concepts is performed by downstream neurons implementing ‘prototype’ decision rule, in which a test example is classified according to the nearest prototype constructed from the few training examples. We show that prototype few-shot learning achieves high few-shot learning accuracy on natural visual concepts using both macaque inferotemporal cortex representations and deep neural network (DNN) models of these representations. We develop a mathematical theory that links few-shot learning to the geometric properties of the neural concept manifolds and demonstrate its agreement with our numerical simulations across different DNNs as well as different layers. Intriguingly, we observe striking mismatches between the geometry of manifolds in intermediate stages of the primate visual pathway and in trained DNNs. Finally, we show that linguistic descriptors of visual concepts can be used to discriminate images belonging to novel concepts, without any prior visual experience of these concepts (a task known as ‘zero-shot’ learning), indicated a remarkable alignment of manifold representations of concepts in visual and language modalities. I will discuss ongoing effort to extend this work to other high level cognitive tasks.

Neural activity in many sensory systems is organized on low-dimensional manifolds by means of convex receptive fields. Neural codes in these areas are constrained by this organization, as not every neural code is compatible with convex receptive fields. The same codes are also constrained by the structure of the underlying neural network. In my talk I will attempt to provide answers to the following natural questions: (i) How do recurrent circuits generate codes that are compatible with the convexity of receptive fields? (ii) How can we utilize the constraints imposed by the convex receptive field to understand the underlying stimulus space. To answer question (i), we describe the combinatorics of the steady states and fixed points of recurrent networks that satisfy the Dale’s law. It turns out the combinatorics of the fixed points are completely determined by two distinct conditions: (a) the connectivity graph of the network and (b) a spectral condition on the synaptic matrix. We give a characterization of exactly which features of connectivity determine the combinatorics of the fixed points. We also find that a generic recurrent network that satisfies Dale's law outputs convex combinatorial codes. To address question (ii), I will describe methods based on ideas from topology and geometry that take advantage of the convex receptive field properties to infer the dimension of (non-linear) neural representations. I will illustrate the first method by inferring basic features of the neural representations in the mouse olfactory bulb.

Talk & Tutorial

Physiological experiments have highlighted how the dendrites of biological neurons can nonlinearly process distributed synaptic inputs. However, it is unclear how qualitative aspects of a dendritic tree, such as its branched morphology, its repetition of presynaptic inputs, voltage-gated ion channels, electrical properties and complex synapses, determine neural computation beyond this apparent nonlinearity. While it has been speculated that the dendritic tree of a neuron can be seen as a multi-layer neural network and it has been shown that such an architecture could be computationally strong, we do not know if that computational strength is preserved under these qualitative biological constraints. Here we simulate multi-layer neural network models of dendritic computation with and without these constraints. We find that dendritic model performance on interesting machine learning tasks is not hurt by most of these constraints and may synergistically benefit from all of them combined. Our results suggest that single real dendritic trees may be able to learn a surprisingly broad range of tasks through the emergent capabilities afforded by their properties.

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.

Several species of insects have been observed to perform accurate path integration, constantly updating a vector memory of their location relative to a starting position, which they can use to take a direct return path. Foraging insects such as bees and ants are also able to store and recall the vectors to return to food locations, and to take novel shortcuts between these locations. Other insects, such as dung beetles, are observed to integrate multimodal directional cues in a manner well described by vector addition. All these processes appear to be functions of the Central Complex, a highly conserved and strongly structured circuit in the insect brain. Modelling this circuit, at the single neuron level, suggests it has general capabilities for vector encoding, vector memory, vector addition and vector rotation that can support a wide range of directed and navigational behaviours.

A major challenge for causal inference from time-series data is the trade-off between computational feasibility and accuracy. Motivated by process motifs for lagged covariance in an autoregressive model with slow mean-reversion, we propose to infer networks of causal relations via pairwise edge measure (PEMs) that one can easily compute from lagged correlation matrices. Motivated by contributions of process motifs to covariance and lagged variance, we formulate two PEMs that correct for confounding factors and for reverse causation. To demonstrate the performance of our PEMs, we consider network interference from simulations of linear stochastic processes, and we show that our proposed PEMs can infer networks accurately and efficiently. Specifically, for slightly autocorrelated time-series data, our approach achieves accuracies higher than or similar to Granger causality, transfer entropy, and convergent crossmapping -- but with much shorter computation time than possible with any of these methods. Our fast and accurate PEMs are easy-to-implement methods for network inference with a clear theoretical underpinning. They provide promising alternatives to current paradigms for the inference of linear models from time-series data, including Granger causality, vector-autoregression, and sparse inverse covariance estimation.

Memory consolidation is the process by which recent experiences are assimilated into long-term memory. In animals, this process requires the offline replay of sequences observed during online exploration in the hippocampus. Recent experimental work has found that salient but task-irrelevant stimuli are systematically excluded from these replay epochs, suggesting that replay samples from an abstracted model of the world, rather than verbatim previous experiences. We find that this phenomenon can be explained parsimoniously and biologically plausibly by a Hebbian spike time-dependent plasticity rule at inhibitory synapses. Using spiking networks at three levels of abstraction–leaky integrate-and-fire, biophysically detailed, and abstract binary–we show that this rule enables efficient inference of a model of the structure of the world. While plasticity has previously mainly been studied at excitatory synapses, we find that plasticity at excitatory synapses alone is insufficient to accomplish this type of structural learning. We present theoretical results in a simplified model showing that in the presence of Hebbian excitatory and inhibitory plasticity, the replayed sequences form a statistical estimator of a latent sequence, which converges asymptotically to the ground truth. Our work outlines a direct link between the synaptic and cognitive levels of memory consolidation, and highlights a potential conceptually distinct role for inhibition in computing with SNNs.

The spike-threshold nonlinearity is a fundamental, yet enigmatic, component of biological computation — despite its role in many theories, it has evaded definitive characterisation. Indeed, much classic work has attempted to limit the focus on spiking by smoothing over the spike threshold or by approximating spiking dynamics with firing-rate dynamics. Here, we take a novel perspective that captures the full potential of spike-based computation. Based on previous studies of the geometry of efficient spike-coding networks, we consider a population of neurons with low-rank connectivity, allowing us to cast each neuron’s threshold as a boundary in a space of population modes, or latent variables. Each neuron divides this latent space into subthreshold and suprathreshold areas. We then demonstrate how a network of inhibitory (I) neurons forms a convex, attracting boundary in the latent coding space, and a network of excitatory (E) neurons forms a concave, repellant boundary. Finally, we show how the combination of the two yields stable dynamics at the crossing of the E and I boundaries, and can be mapped onto a constrained optimization problem. The resultant EI networks are balanced, inhibition-stabilized, and exhibit asynchronous irregular activity, thereby closely resembling cortical networks of the brain. Moreover, we demonstrate how such networks can be tuned to either suppress or amplify noise, and how the composition of inhibitory convex and excitatory concave boundaries can result in universal function approximation. Our work puts forth a new theory of biologically-plausible computation in balanced spiking networks, and could serve as a novel framework for scalable and interpretable computation with spikes.

A central problem in biological learning is how information about the outcome of a decision or behavior can be used to reliably guide learning across distributed neural circuits while obeying biological constraints. This “credit assignment” problem is commonly solved in artificial neural networks through supervised gradient descent and the backpropagation algorithm. In contrast, biological learning is typically modelled using unsupervised Hebbian learning rules. While these rules only use local information to update synaptic weights, and are sometimes combined with weight constraints to reflect a diversity of excitatory (only positive weights) and inhibitory (only negative weights) cell types, they do not prescribe a clear mechanism for how to coordinate learning across multiple layers and propagate error information accurately across the network. In recent years, several groups have drawn inspiration from the known dendritic non-linearities of pyramidal neurons to propose new learning rules and network architectures that enable biologically plausible multi-layer learning by processing error information in segregated dendrites. Meanwhile, recent experimental results from the hippocampus have revealed a new form of plasticity—Behavioral Timescale Synaptic Plasticity (BTSP)—in which large dendritic depolarizations rapidly reshape synaptic weights and stimulus selectivity with as little as a single stimulus presentation (“one-shot learning”). Here we explore the implications of this new learning rule through a biologically plausible implementation in a rate neuron network. We demonstrate that regulation of dendritic spiking and BTSP by top-down feedback signals can effectively coordinate plasticity across multiple network layers in a simple pattern recognition task. By analyzing hidden feature representations and weight trajectories during learning, we show the differences between networks trained with standard backpropagation, Hebbian learning rules, and BTSP.

The brain efficiently performs nonlinear computations through its intricate networks of spiking neurons, but how this is done remains elusive. While recurrent spiking networks implementing linear computations can be directly derived and easily understood (e.g., in the spike coding network (SCN) framework), the connectivity required for nonlinear computations can be harder to interpret, as they require additional non-linearities (e.g., dendritic or synaptic) weighted through supervised training. Here we extend the SCN framework to directly implement any polynomial dynamical system. This results in networks requiring multiplicative synapses, which we term the multiplicative spike coding network (mSCN). We demonstrate how the required connectivity for several nonlinear dynamical systems can be directly derived and implemented in mSCNs, without training. We also show how to precisely carry out higher-order polynomials with coupled networks that use only pair-wise multiplicative synapses, and provide expected numbers of connections for each synapse type. Overall, our work provides an alternative method for implementing nonlinear computations in spiking neural networks, while keeping all the attractive features of standard SCNs such as robustness, irregular and sparse firing, and interpretable connectivity. Finally, we discuss the biological plausibility of mSCNs, and how the high accuracy and robustness of the approach may be of interest for neuromorphic computing.

Drosophila olfactory sensory hairs ("sensilla") typically house two olfactory receptor neurons (ORNs) which can laterally inhibit each other via electrical ("ephaptic") coupling. ORN pairing is highly stereotyped and genetically determined. Thus, olfactory signals arriving in the Antennal Lobe (AL) have been pre-processed by a fixed and shallow network at the periphery. To uncover the functional significance of this organization, we developed a nonlinear phenomenological model of asymmetrically coupled ORNs responding to odor mixture stimuli. We derived an analytical solution to the ORNs’ dynamics, which shows that the peripheral network can extract the valence of specific odor mixtures via transient amplification. Our model predicts that for efficient read-out of the amplified valence signal there must exist specific patterns of downstream connectivity that reflect the organization at the periphery. Analysis of AL→Lateral Horn (LH) fly connectomic data reveals evidence directly supporting this prediction. We further studied the effect of ephaptic coupling on olfactory processing in the AL→Mushroom Body (MB) pathway. We show that stereotyped ephaptic interactions between ORNs lead to a clustered odor representation of glomerular responses. Such clustering in the AL is an essential assumption of theoretical studies on odor recognition in the MB. Together our work shows that preprocessing of olfactory stimuli by a fixed and shallow network increases sensitivity to specific odor mixtures, and aids in the learning of novel olfactory stimuli. Work led by Palka Puri, in collaboration with Chih-Ying Su and Shiuan-Tze Wu.