Lai-Sang Young’s research while affiliated with New York University and other places

C. elegans locomotion is composed of switches between forward and reversal states punctuated by turns. This locomotory capability is necessary for the nematode to move towards attractive stimuli, escape noxious chemicals, and explore its environment. Although experimentalists have identified a number of premotor neurons as drivers of forward and reverse motion, how these neurons work together to produce the behaviors observed remains to be understood. Towards a better understanding of C. elegans neurodynamics, we present in this paper a minimally parameterized, biophysical dynamical systems model of the premotor network. Our model consists of a recurrently connected collection of premotor neurons (the core group) driven by over a hundred sensory and interneurons that provide diverse feedforward inputs to the core group. It is data-driven in the sense that the choice of neurons in the core group follows experimental guidance, anatomical structures are dictated by the connectome, and physiological parameters are deduced from whole-brain imaging and voltage clamps data. When simulated with realistic input signals, our model produces premotor activity that closely resembles experimental data: from the seemingly random switching between forward and reversal behaviors to the synchronization of subnetworks to various higher-order statistics. We posit that different roles are played by gap junctions and synaptic connections in switching dynamics. The model correctly predicts behavior such as dwelling versus roaming as a result of the synaptic inputs received, and we demonstrate that it can be used to study how the activity level of certain individual neurons impacts behavior.

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This is supplementary materials for Xiao et al 2024

Biologically detailed models of brain circuitry are challenging to build and simulate due to the large number of neurons, their complex interactions, and the many unknown physiological parameters. Simplified mathematical models are more tractable, but harder to evaluate when too far removed from neuroanatomy/physiology. We propose that a multiscale model, coarse-grained (CG) while preserving local biological details, offers the best balance between biological realism and computability. This paper presents such a model. Generally, CG models focus on the interaction between groups of neurons—here termed “pixels”—rather than individual cells. In our case, dynamics are alternately updated at intra- and interpixel scales, with one informing the other, until convergence to equilibrium is achieved on both scales. An innovation is how we exploit the underlying biology: Taking advantage of the similarity in local anatomical structures across large regions of the cortex, we model intrapixel dynamics as a single dynamical system driven by “external” inputs. These inputs vary with events external to the pixel, but their ranges can be estimated a priori . Precomputing and tabulating all potential local responses speed up the updating procedure significantly compared to direct multiscale simulation. We illustrate our methodology using a model of the primate visual cortex. Except for local neuron-to-neuron variability (necessarily lost in any CG approximation) our model reproduces various features of large-scale network models at a tiny fraction of the computational cost. These include neuronal responses as a consequence of their orientation selectivity, a primary function of visual neurons.

The workshop aimed to explore the use of new mathematical and computational approaches to investigate the fundamental principles governing the organization and dynamics of biological systems. This necessitated conversations among mathematical biologists working at different scales, from molecular to organismal levels. The meeting aimed to encourage interdisciplinary collaborations and showcase recent advances in diverse areas.

This paper is about a class of stochastic reaction networks. Of interest are the dynamics of interconversion among a finite number of substances through reactions that consume some of the substances and produce others. The models we consider are continuous-time Markov jump processes, intended as idealizations of a broad class of biological networks. Reaction rates depend linearly on “enzymes,” which are among the substances produced, and a reaction can occur only in the presence of sufficient upstream material. We present rigorous results for this class of stochastic dynamical systems, the mean-field behaviors of which are described by ordinary differential equations (ODEs). Under the assumption of exponential network growth, we identify certain ODE solutions as being potentially traceable and give conditions on network trajectories which, when rescaled, can with high probability be approximated by these ODE solutions. This leads to a complete characterization of the ω -limit sets of such network solutions (as points or random tori). Dimension reduction is noted depending on the number of enzymes. The second half of this paper is focused on depletion dynamics, i.e., dynamics subsequent to the “phase transition” that occurs when one of the substances becomes unavailable. The picture can be complex, for the depleted substance can be produced intermittently through other network reactions. Treating the model as a slow–fast system, we offer a mean-field description, a first step to understanding what we believe is one of the most natural bifurcations for reaction networks.

Key features of biological activity can often be captured by transitions between a finite number of semi-stable states that correspond to behaviors or decisions. We present here a broad class of dynamical systems that are ideal for modeling such activity. The models we propose are chaotic heteroclinic networks with nontrivial intersections of stable and unstable manifolds. Due to the sensitive dependence on initial conditions, transitions between states are seemingly random. Dwell times, exit distributions, and other transition statistics can be built into the model through geometric design and can be controlled by tunable parameters. To test our model’s ability to simulate realistic biological phenomena, we turned to one of the most studied organisms, C. elegans, well known for its limited behavioral states. We reconstructed experimental data from two laboratories, demonstrating the model’s ability to quantitatively reproduce dwell times and transition statistics under a variety of conditions. Stochastic switching between dominant states in complex dynamical systems has been extensively studied and is often modeled as Markov chains. As an alternative, we propose here a new paradigm, namely, chaotic heteroclinic networks generated by deterministic rules (without the necessity for noise). Chaotic heteroclinic networks can be used to model systems with arbitrary architecture and size without a commensurate increase in phase dimension. They are highly flexible and able to capture a wide range of transition characteristics that can be adjusted through control parameters.

Key features of biological activity can often be captured by transitions between a finite number of semi-stable states that correspond to behaviors or decisions. We present here a broad class of dynamical systems that are ideal for modeling such activity. The models we propose are chaotic heteroclinic networks with nontrivial intersections of stable and unstable manifolds. Due to the sensitive dependence on initial conditions, transitions between states are seemingly random. Dwell times, exit distributions, and other transition statistics can be built into the model through geometric design and can be controlled by tunable parameters. To test our model's ability to simulate realistic biological phenomena, we turned to one of the most studied organisms, {\it C. elegans}, well known for its limited behavioral states. We reconstructed experimental data from two laboratories, demonstrating the model's ability to quantitatively reproduce dwell times and transition statistics under a variety of conditions. Stochastic switching between dominant states in complex dynamical systems has been extensively studied and is often modeled as Markov chains. As an alternative, we propose here a new paradigm, namely chaotic heteroclinic networks generated by deterministic rules (without the necessity for noise). Chaotic heteroclinic networks can be used to model systems with arbitrary architecture and size without a commensurate increase in phase dimension. They are highly flexible and able to capture a wide range of transition characteristics that can be adjusted through control parameters.

The brain produces rhythms in a variety of frequency bands. Some are likely by-products of neuronal processes; others are thought to be top-down. Produced entirely naturally, these rhythms have clearly recognizable beats, but they are very far from periodic in the sense of mathematics. The signals are broad-band, episodic, wandering in amplitude and frequency; the rhythm comes and goes, degrading and regenerating. Gamma rhythms, in particular, have been studied by many authors in computational neuroscience, using reduced models as well as networks of hundreds to thousands of integrate-and-fire neurons. All of these models captured successfully the oscillatory nature of gamma rhythms, but the irregular character of gamma in reduced models has not been investigated thoroughly. In this article, we tackle the mathematical question of whether signals with the properties of brain rhythms can be generated from low dimensional dynamical systems. We found that while adding white noise to single periodic cycles can to some degree simulate gamma dynamics, such models tend to be limited in their ability to capture the range of behaviors observed. Using an ODE with two variables inspired by the FitzHugh-Nagumo and Leslie-Gower models, with stochastically varying coefficients designed to control independently amplitude, frequency, and degree of degeneracy, we were able to replicate the qualitative characteristics of natural brain rhythms. To demonstrate model versatility, we simulate the power spectral densities of gamma rhythms in various brain states recorded in experiments.

This paper is about neural mechanisms of direction selectivity (DS) in Macaque primary visual cortex, V1. We present data (on male macaque) showing strong DS in a majority of simple cells in V1 layer 4Cα, the cortical layer that receives direct afferent input from the Magnocellular division of the Lateral Geniculate Nucleus (LGN). Magnocellular LGN cells are not direction-selective. To understand the mechanisms of DS, we built a large-scale, recurrent model of spiking neurons called DSV1. Like its predecessors, DSV1 reproduces many visual response properties of V1 cells including orientation selectivity. Two important new features of DSV1 are (a) DS is initiated by small, consistent dynamic differences in the visual responses of OFF and ON Magnocellular LGN cells, and (b) DS in the responses of most model simple cells is increased over those of their feedforward inputs; this increase is achieved through dynamic interaction of feedforward and intra-cortical synaptic currents without the use of intra-cortical direction-specific connections. The DSV1 model emulates experimental data in the following ways: 1) most 4Cα Simple cells were highly direction-selective but 4Cα Complex cells were not; 2) the preferred directions of the model's direction-selective Simple cells were invariant with spatial and temporal frequency; 3) the distribution of the Preferred/Opposite ratio across the model's population of cells was very close to that found in experiments. The strong quantitative agreement between DS in data and in model simulations suggests that the neural mechanisms of DS in DSV1 may be similar to those in the real visual cortex.SIGNIFICANCE STATEMENT:Motion perception is a vital part of our visual experience of the world. In monkeys, whose vision resembles that of humans, the neural computation of the direction of a moving target starts in the primary visual cortex, V1, in layer 4Cα that receives input from the eye through the Lateral Geniculate Nucleus (LGN). How Direction-Selectivity (DS) is generated in layer 4Cα is an outstanding unsolved problem in theoretical neuroscience. In this paper, we offer a solution based on plausible biological mechanisms: We present a new large-scale circuit model in which DS originates from slightly different LGN ON/OFF response time-courses and is enhanced in cortex without the need for direction-specific intra-cortical connections. The model's DS is in quantitative agreement with experiments.