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C. eleganslocomotion 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 ofC. elegansneurodynamics, we present in this paper a minimally parameterized, biology-based 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. Using the model we identify signal neurons that strongly influence switches between behavioral states and core neurons that play an important role in integrating signal information. The model produces switching statistics that underlie behaviors such as dwelling versus roaming as a result of the synaptic inputs received. 

Abstract We investigate structural features and processes associated with the onset of systemic conflict using an approach which integrates complex systems theory with network modeling and analysis. We present a signed network model of cooperation and conflict dynamics in the context of international relations between states. The model evolves ties between nodes under the influence of a structural balance force and a dyad-specific force. Model simulations exhibit a sharp bifurcation from peace to systemic war as structural balance pressures increase, a bistable regime in which both peace and war stable equilibria exist, and a hysteretic reverse bifurcation from war to peace. We show how the analytical expression we derive for the peace-to-war bifurcation condition implies that polarized network structure increases susceptibility to systemic war. We develop a framework for identifying patterns of relationship perturbations that are most destabilizing and apply it to the network of European great powers before World War I. We also show that the model exhibits critical slowing down, in which perturbations to the peace equilibrium take longer to decay as the system draws closer to the bifurcation. We discuss how our results relate to international relations theories on the causes and catalysts of systemic war. 

Nonlinear ODEs can rarely be solved analytically. Koopman operator theory provides a way to solve two-dimensional nonlinear systems, under suitable restrictions, by mapping nonlinear dynamics to a linear space using Koopman eigenfunctions. Unfortunately, finding such eigenfunctions is difficult. We introduce a method for constructing Koopman eigenfunctions from a two-dimensional nonlinear ODE’s one-dimensional invariant manifolds. This method, when successful, allows us to find analytical solutions for autonomous, nonlinear systems. Previous data-driven methods have used Koopman theory to construct local Koopman eigenfunction approximations valid in different regions of phase space; our method finds analytic Koopman eigenfunctions that are exact and globally valid. We demonstrate our Koopman method of solving nonlinear systems on one-dimensional and two-dimensional ODEs. The nonlinear examples considered have simple expressions for their codimension-1 invariant manifolds which produce tractable analytical solutions. Thus our method allows for the construction of analytical solutions for previously unsolved ODEs. It also highlights the connection between invariant manifolds and eigenfunctions in nonlinear ODEs and presents avenues for extending this method to solve more nonlinear systems. 

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.