Equivalent Electronic Circuit of a System of Oscillators Connected With Periodically Variable Stiffness

In this paper, we have shown the electronic circuit equivalence of a mechanical system consists of two oscillators coupled with each other. The mechanical design has the effects of the magnetic, resistance forces and the spring constant of the system is periodically varying. We have shown that the system’s state variables, such as the displacements and the velocities, under the effects of different forces, lead to some nonlinear behaviors, like a transition from the fixed point attractor to the chaotic attractor through the periodic and quasi-periodic attractors. We have constructed the equivalent electronic circuit of this mechanical system and have verified the numerically obtained behaviors using the electronic circuit.

Hidden Hypergraphs, Error-Correcting Codes, and Critical Learning in Hopfield Networks

In 1943, McCulloch and Pitts introduced a discrete recurrent neural network as a model for computation in brains. The work inspired breakthroughs such as the first computer design and the theory of finite automata. We focus on learning in Hopfield networks, a special case with symmetric weights and fixed-point attractor dynamics. Specifically, we explore minimum energy flow (MEF) as a scalable convex objective for determining network parameters. We catalog various properties of MEF, such as biological plausibility, and them compare to classical approaches in the theory of learning. Trained Hopfield networks can perform unsupervised clustering and define novel error-correcting coding schemes. They also efficiently find hidden structures (cliques) in graph theory. We extend this known connection from graphs to hypergraphs and discover n-node networks with robust storage of 2Ω(n1−ϵ) memories for any ϵ>0. In the case of graphs, we also determine a critical ratio of training samples at which networks generalize completely.

Modelling the impact of structural directionality on connectome-based models of neural activity

Understanding structure--function relationships in the brain remains an important challenge in neuroscience. However, whilst structural brain networks are intrinsically directed, due to the prevalence of chemical synapses in the cortex, most studies in network neuroscience represent the brain as an undirected network. Here, we explore the role that directionality plays in shaping transition dynamics of functional brain states. Using a system of Hopfield neural elements with heterogeneous structural connectivity given by different species and parcellations (cat, Caenorhabditis elegans and two macaque networks), we investigate the effect of removing directionality of connections on brain capacity, which we quantify via its ability to store attractor states. In addition to determining large numbers of fixed-point attractor sets, we deploy the recently developed basin stability technique in order to assess the global stability of such brain states, which can be considered a proxy for network state robustness. Our study indicates that not only can directed network topology have a significant effect on the information capacity of connectome-based networks, but it can also impact significantly the domains of attraction of the aforementioned brain states. In particular, we find network modularity to be a key mechanism underlying the formation of neural activity patterns, and moreover, our results suggest that neglecting network directionality has the scope to eliminate states that correlate highly with the directed modular structure of the brain. A numerical analysis of the distribution of attractor states identified a small set of prototypical direction-dependent activity patterns that potentially constitute a `skeleton' of the non-stationary dynamics typically observed in the brain. This study thereby emphasizes the substantial role network directionality can have in shaping the brain's ability to both store and process information.

Solitonic Fixed Point Attractors in the Complex Ginzburg–Landau Equation for Associative Memories

It was recently shown that the nonlinear Schrodinger equation with a simplified dissipative perturbation features a zero-velocity solitonic solution of non-zero amplitude which can be used in analogy to attractors of Hopfield’s associative memory. In this work, we consider a more complex dissipative perturbation adding the effect of two-photon absorption and the quintic gain/loss effects that yields the complex Ginzburg–Landau equation (CGLE). We construct a perturbation theory for the CGLE with a small dissipative perturbation, define the behavior of the solitonic solutions with parameters of the system and compare the solution with numerical simulations of the CGLE. We show, in a similar way to the nonlinear Schrodinger equation with a simplified dissipation term, a zero-velocity solitonic solution of non-zero amplitude appears as an attractor for the CGLE. In this case, the amplitude and velocity of the solitonic fixed point attractor does not depend on the quintic gain/loss effects. Furthermore, the effect of two-photon absorption leads to an increase in the strength of the solitonic fixed point attractor.

Robust computation with rhythmic spike patterns

Information coding by precise timing of spikes can be faster and more energy efficient than traditional rate coding. However, spike-timing codes are often brittle, which has limited their use in theoretical neuroscience and computing applications. Here, we propose a type of attractor neural network in complex state space and show how it can be leveraged to construct spiking neural networks with robust computational properties through a phase-to-timing mapping. Building on Hebbian neural associative memories, like Hopfield networks, we first propose threshold phasor associative memory (TPAM) networks. Complex phasor patterns whose components can assume continuous-valued phase angles and binary magnitudes can be stored and retrieved as stable fixed points in the network dynamics. TPAM achieves high memory capacity when storing sparse phasor patterns, and we derive the energy function that governs its fixed-point attractor dynamics. Second, we construct 2 spiking neural networks to approximate the complex algebraic computations in TPAM, a reductionist model with resonate-and-fire neurons and a biologically plausible network of integrate-and-fire neurons with synaptic delays and recurrently connected inhibitory interneurons. The fixed points of TPAM correspond to stable periodic states of precisely timed spiking activity that are robust to perturbation. The link established between rhythmic firing patterns and complex attractor dynamics has implications for the interpretation of spike patterns seen in neuroscience and can serve as a framework for computation in emerging neuromorphic devices.

Verification of Continuous Time Recurrent Neural Networks (Benchmark Proposal)

This manuscript presents a description and implementation of two benchmark problems for continuous-time recurrent neural network (RNN) verification. The first problem deals with the approximation of a vector field for a fixed point attractor located at the origin, whereas the second problem deals with the system identification of a forced damped pendulum. While the verification of neural networks is complicated and often impenetrable to the majority of verification techniques, continuous-time RNNs represent a class of networks that may be accessible to reachability methods for nonlinear ordinary differential equations (ODEs) derived originally in biology and neuroscience. Thus, an understanding of the behavior of a RNN may be gained by simulating the nonlinear equations from a diverse set of initial conditions and inputs, or considering reachability analysis from a set of initial conditions. The verification of continuous-time RNNs is a research area that has received little attention and if the research community can achieve meaningful results in this domain, then this class of neural networks may prove to be a superior approach in solving complex problems compared to other network architectures.

Size segregation of intruders in perpetual granular avalanches

Granular flows such as landslides, debris flows and avalanches are systems of particles with a large range of particle sizes that typically segregate while flowing. The physical mechanisms responsible for this process, however, are still poorly understood, and there is no predictive framework for ascertaining the segregation behaviour of a given system of particles. Here, we provide experimental evidence of individual large intruder particles being attracted to a fixed point in a dry two-dimensional flow of particles of otherwise uniform size. A continuum theory is proposed which captures this effect using only a single fitting parameter that describes the rate of segregation, given knowledge of the bulk flow field. Predictions of the continuum theory are compared with the experimental findings, both for the typical location and velocity field of a range of intruder sizes. For large intruder particle sizes, the continuum model successfully predicts that a fixed point attractor will form, where intruders are drawn to a single location.

A mathematical model relates intracellular TLR4 oscillations to sepsis progression

Oscillations drive many biological processes and their modulation is determinant for various pathologies. In sepsis syndrome, Toll-like receptor 4 (TLR4) is a key sensor for signaling the presence of Gram-negative bacteria. Its expression and activity, along with its intracellular trafficking rates shift the equilibrium between the pro‐ and anti-inflammatory downstream signaling cascades, leading to either the physiological resolution of the bacterial stimulation or to sepsis. We hypothesize that the initial tlr4 expression in patients diagnosed with sepsis and TLR4 dynamic concentration changes on the cell membrane or intracellularly, dictates how the sepsis syndrome is initiated. Using a set of three differential equations, we defined the TLR4 flux between relevant cell organelles. We obtained three different regions in the phase space: 1. a limit-cycle describing unstimulated physiological oscillations, 2. a fixed-point attractor resulting from moderate LPS stimulation that is resolved and 3. a double-attractor resulting from sustained LPS stimulation that leads to sepsis. We tested the models against hospital data of sepsis patients and we correctly evaluate the clinical outcome of these patients.