Search

Identifying cliques in dense networks remains a formidable challenge, even with significant advances in computational power and methodologies. To tackle this, numerous algorithms have been developed to optimize time and memory usage, implemented across diverse programming languages. Yet, the inherent NP-completeness of the problem continues to hinder performance on large-scale networks, often resulting in memory leaks and slow computations. In the present study, we critically evaluate classic algorithms to pinpoint computational bottlenecks and introduce novel set-theoretical approaches tailored for network clique computation. Our proposed algorithms are rigorously implemented and benchmarked against existing Python-based solutions, demonstrating superior performance. These findings underscore the potential of set-theoretical techniques to drive substantial performance gains in network analysis.

We construct systematically experimental steady-state bifurcation diagrams for entorhinal cortex neurons. A slowly ramped voltage-clamp electrophysiology protocol serves as closed-loop feedback controlled experiment for the subsequent current-clamp open-loop protocol on the same cell. In this way, the voltage-clamped experiment determines dynamically stable and unstable (hidden) steady states of the current-clamp experiment. The transitions between observable steady states and observable spiking states in the current-clamp experiment reveal stability and bifurcations of the steady states, completing the steady-state bifurcation diagram.

Synchronization is a widespread phenomenon in the brain. Despite numerous studies, the specific parameter configurations of the synaptic network structure and learning rules needed to achieve robust and enduring synchronization in neurons driven by spike-timing-dependent plasticity (STDP) and temporal networks subject to homeostatic structural plasticity (HSP) rules remain unclear. Here, we bridge this gap by determining the configurations required to achieve high and stable degrees of complete synchronization (CS) and phase synchronization (PS) in time-varying small-world and random neural networks driven by STDP and HSP. In particular, we found that decreasing $P$ (which enhances the strengthening effect of STDP on the average synaptic weight) and increasing $F$ (which speeds up the swapping rate of synapses between neurons) always lead to higher and more stable degrees of CS and PS in small-world and random networks, provided that the network parameters such as the synaptic time delay $\tau_c$, the average degree $\langle k \rangle$, and the rewiring probability $\beta$ have some appropriate values. When $\tau_c$, $\langle k \rangle$, and $\beta$ are not fixed at these appropriate values, the degree and stability of CS and PS may increase or decrease when $F$ increases, depending on the network topology. It is also found that the time delay $\tau_c$ can induce intermittent CS and PS whose occurrence is independent $F$. Our results could have applications in designing neuromorphic circuits for optimal information processing and transmission via synchronization phenomena., Comment: 28 pages, 86 references, 8 figures, 2 Tables

We propose a generalization of the neurotransmitter release model proposed in \emph{Rodrigues et al. (PNAS, 2016)}. We increase the complexity of the underlying slow-fast system by considering a degree-four polynomial as parametrization of the critical manifold. We focus on the possible transient and asymptotic dynamics, exploiting the so-called entry-exit function to describe slow parts of the dynamics. We provide extensive numerical simulations, complemented by numerical bifurcation analysis., Comment: 19 pages, 31 figures

From the action potentials of neurons and cardiac cells to the amplification of calcium signals in oocytes, excitability is a hallmark of many biological signalling processes. In recent years, excitability in single cells has been related to multiple-timescale dynamics through canards, special solutions which determine the effective thresholds of the all-or-none responses. However, the emergence of excitability in large populations remains an open problem. Here, we show that the mechanism of excitability in large networks and mean-field descriptions of coupled quadratic integrate-and-fire (QIF) cells mirrors that of the individual components. We initially exploit the Ott-Antonsen ansatz to derive low-dimensional dynamics for the coupled network and use it to describe the structure of canards via slow periodic forcing. We demonstrate that the thresholds for onset and offset of population firing can be found in the same way as those of the single cell. We combine theoretical analysis and numerical computations to develop a novel and comprehensive framework for excitability in large populations, applicable not only to models amenable to Ott-Antonsen reduction, but also to networks without a closed-form mean-field limit, in particular sparse networks.

The phenomenon of slow passage through a Hopf bifurcation is ubiquitous in multiple-timescale dynamical systems, where a slowly-varying quantity replacing a static parameter induces the solutions of the resulting slow-fast system to feel the effect of a Hopf bifurcation with a delay. This phenomenon is well understood in the context of smooth slow-fast dynamical systems. In the present work, we study for the first time this phenomenon in piecewise linear (PWL) slow-fast systems. This special class of systems is indeed known to reproduce all features of their smooth counterpart while being more amenable to quantitative analysis and offering some level of simplification, in particular through the existence of canonical (linear) slow manifolds. We provide conditions for a PWL slow-fast system to exhibit a slow passage through a Hopf-like bifurcation, in link with the number of linearity zones considered in the system and possible connections between canonical attracting and repelling slow manifolds. In doing so, we fully describe the so-called way-in/way-out function. Finally, we investigate this slow passage effect in the Doi-Kumagai model, a neuronal PWL model exhibiting elliptic bursting oscillations.

We report a detailed analysis on the emergence of bursting in a recently developed neural mass model that takes short-term synaptic plasticity into account. The one being used here is particularly important, as it represents an exact meanfield limit of synaptically coupled quadratic integrate & fire neurons, a canonical model for type I excitability. In absence of synaptic dynamics, a periodic external current with a slow frequency {\epsilon} can lead to burst-like dynamics. The firing patterns can be understood using techniques of singular perturbation theory, specifically slow-fast dissection. In the model with synaptic dynamics the separation of timescales leads to a variety of slow-fast phenomena and their role for bursting is rendered inordinately more intricate. Canards are one of the main slow-fast elements on the route to bursting. They describe trajectories evolving nearby otherwise repelling locally invariant sets of the system and are found in the transition region from subthreshold dynamics to bursting. For values of the timescale separation nearby the singular limit {\epsilon} = 0, we report peculiar jump-on canards, which block a continuous transition to bursting. In the biologically more plausible regime of {\epsilon} this transition becomes continuous and bursts emerge via consecutive spike-adding transitions. The onset of bursting is of complex nature and involves mixed-type like torus canards, which form the very first spikes of the burst and revolve nearby fast-subsystem repelling limit cycles. We provide numerical evidence for the same mechanisms to be responsible for the emergence of bursting in the quadratic integrate & fire network with plastic synapses. The main conclusions apply for the network, owing to the exactness of the meanfield limit.

We study a class of planar integrate and fire (IF) models called adaptive integrate and fire (AIF) models, which possesses an adaptation variable on top of membrane potential, and whose subthreshold dynamics is piecewise linear (PWL). These AIF models therefore have two reset conditions, which enable bursting dynamics to emerge for suitable parameter values. Such models can be thought of as hybrid dynamical systems. We consider a particular slow dynamics within AIF models and prove the existence of bursting cycles with $N$ resets, for any integer $N$. Furthermore, we study the transition between $N$- and $(N+1)$-reset cycles upon vanishingly small parameter variations and prove (for $N=2$) that such transitions are organised by canard cycles. Finally, using numerical continuation we compute branches of bursting cycles, including canard-explosive branches, in these AIF models, by suitably recasting the periodic problem as a two-point boundary-value problem.

We revisit elliptic bursting dynamics from the viewpoint of torus canard solutions. We show that at the transition to and from elliptic burstings, classical or mixed-type torus canards can appear, the difference between the two being the fast subsystem bifurcation that they approach, saddle-node of cycles for the former and subcritical Hopf for the latter. We first showcase such dynamics in a Wilson-Cowan type elliptic bursting model, then we consider minimal models for elliptic bursters in view of finding transitions to and from bursting solutions via both kinds of torus canards. We first consider the canonical model proposed by Izhikevich (ref. [22] in the manuscript) and adapted to elliptic bursting by Ju, Neiman, Shilnikov (ref. [24] in the manuscript), and we show that it does not produce mixed-type torus canards due to a nongeneric transition at one end of the bursting regime. We therefore introduce a perturbative term in the slow equation, which extends this canonical form to a new one that we call Leidenator and which supports the right transitions to and from elliptic bursting via classical and mixed-type torus canards, respectively. Throughout the study, we use singular flows ($\varepsilon=0$) to predict the full system's dynamics ($\varepsilon>0$ small enough). We consider three singular flows: slow, fast and average slow, so as to appropriately construct singular orbits corresponding to all relevant dynamics pertaining to elliptic bursting and torus canards.

Living systems are subject to the arrow of time; from birth, they undergo complex transformations (self-organization) in a constant battle for survival, but inevitably ageing and disease trap them to death. Can ageing be understood and eventually reversed? What tools can be employed to further our understanding of ageing? The present article is an invitation for biologists and clinicians to consider key conceptual ideas and computational tools (known to mathematicians and physicists), which potentially may help dissect some of the underlying processes of ageing and disease. Specifically, we first discuss how to classify and analyze complex systems, as well as highlight critical theoretical difficulties that make complex systems hard to study. Subsequently, we introduce Topological Data Analysis - a novel Big Data tool - which may help in the study of complex systems since it extracts knowledge from data in a holistic approach via topological considerations. These conceptual ideas and tools are discussed in a rather informal way to pave future discussions and collaborations between mathematicians and biologists studying ageing.

The mathematical classification of complex bursting oscillations in multiscale excitable systems, seen for example in physics and neuroscience, has been the subject of active enquiry since the early 1980s. This classification problem is fundamental as it also establishes analytical and numerical foundations for studying complex temporal behaviours in multiple timescale models. This manuscript begins by reviewing the seminal works of Rinzel and Izhikevich in classifying bursting patterns of excitable cell models. Moreover, we recall an alternative, yet complementary, mathematical classification approach by Golubitsky, which together with the Rinzel-Izhikevich proposals provide the state-of-the-art foundations to the classification problem. Unexpectedly, while keeping within the Rinzel-Izhikevich framework, we find novel cases of bursting mechanisms not considered before. Moving beyond the state-of-the-art, we identify novel bursting mechanisms that fall outside the current classifications. This leads us towards a new classification, which requires the analysis of both the fast and the slow subsystems of an underlying slow-fast model. This new classification allows the dynamical dissection of a larger class of bursters. To substantiate this, we add a new class of bursters with at least two slow variables, which we denote folded-node bursters, to convey the idea that the bursts are initiated or annihilated via a folded-node singularity. In fact, there are two main families of folded-node bursters, depending upon the phase of the bursting cycle during which folded-node dynamics occurs. If it occurs during the silent phase, we obtain the classical folded-node bursting (or simply folded-node bursting). If it occurs during the active phase, we have cyclic folded-node bursting. We classify both families and give examples of minimal systems displaying these novel types of bursting behaviour., Comment: 29 pages, 14 figures, 2 tables

We present a rigorous framework for the local analysis of canards and slow passages through bifurcations in a wide class of infinite-dimensional dynamical systems with time-scale separation. The framework is applicable to models where an infinite-dimensional dynamical system for the fast variables is coupled to a finite-dimensional dynamical system for slow variables. We prove the existence of centre-manifolds for generic models of this type, and study the reduced, finite-dimensional dynamics near bifurcations of (possibly) patterned steady states in the layer problem. Theoretical results are complemented with detailed examples and numerical simulations covering systems of local- and nonlocal-reaction diffusion equations, neural field models, and delay-differential equations. We provide analytical foundations for numerical observations recently reported in literature, such as spatio-temporal canards and slow-passages through Hopf bifurcations in spatially-extended systems subject to slow parameter variations. We also provide a theoretical analysis of slow passage through a Turing bifurcation in local and nonlocal models.

Canards are special solutions to ordinary differential equations that follow invariant repelling slow manifolds for long time intervals. In realistic biophysical single cell models, canards are responsible for several complex neural rhythms observed experimentally, but their existence and role in spatially-extended systems is largely unexplored. We describe a novel type of coherent structure in which a spatial pattern displays temporal canard behaviour. Using interfacial dynamics and geometric singular perturbation theory, we classify spatio-temporal canards and give conditions for the existence of folded-saddle and folded-node canards. We find that spatio-temporal canards are robust to changes in the synaptic connectivity and firing rate. The theory correctly predicts the existence of spatio-temporal canards with octahedral symmetries in a neural field model posed on the unit sphere.

Computation is classically studied in terms of automata, formal languages and algorithms; yet, the relation between neural dynamics and symbolic representations and operations is still unclear in traditional eliminative connectionism. Therefore, we suggest a unique perspective on this central issue, to which we would like to refer as to transparent connectionism, by proposing accounts of how symbolic computation can be implemented in neural substrates. In this study we first introduce a new model of dynamics on a symbolic space, the versatile shift, showing that it supports the real-time simulation of a range of automata. We then show that the Goedelization of versatile shifts defines nonlinear dynamical automata, dynamical systems evolving on a vectorial space. Finally, we present a mapping between nonlinear dynamical automata and recurrent artificial neural networks. The mapping defines an architecture characterized by its granular modularity, where data, symbolic operations and their control are not only distinguishable in activation space, but also spatially localizable in the network itself, while maintaining a distributed encoding of symbolic representations. The resulting networks simulate automata in real-time and are programmed directly, in absence of network training. To discuss the unique characteristics of the architecture and their consequences, we present two examples: i) the design of a Central Pattern Generator from a finite-state locomotive controller, and ii) the creation of a network simulating a system of interactive automata that supports the parsing of garden-path sentences as investigated in psycholinguistics experiments.

Canard-induced phenomena have been extensively studied in the last three decades, both from the mathematical and from the application viewpoints. Canards in slow-fast systems with (at least) two slow variables, especially near folded-node singularities, give an essential generating mechanism for Mixed-Mode oscillations (MMOs) in the framework of smooth multiple timescale systems. There is a wealth of literature on such slow-fast dynamical systems and many models displaying canard-induced MMOs, in particular in neuroscience. In parallel, since the late 1990s several papers have shown that the canard phenomenon can be faithfully reproduced with piecewise-linear (PWL) systems in two dimensions although very few results are available in the three-dimensional case. The present paper aims to bridge this gap by analysing canonical PWL systems that display folded singularities, primary and secondary canards, with a similar control of the maximal winding number as in the smooth case. We also show that the singular phase portraits are compatible in both frameworks. Finally, we show on an example how to construct a (linear) global return and obtain robust PWL MMOs., Comment: to appear in SIAM Review (2016)

A subcritical pattern-forming system with nonlinear advection in a bounded domain is recast as a slow-fast system in space and studied using a combination of geometric singular perturbation theory and numerical continuation. Two types of solutions describing the possible location of stationary fronts are identified, whose origin is traced to the onset of convective and absolute instability when the system is unbounded. The former are present only for nonzero upstream boundary conditions and provide a quantitative understanding of noise-sustained structures in systems of this type. The latter correspond to the onset of a global mode and are present even with zero upstream boundary condition. The role of canard trajectories in the nonlinear transition between these states is clarified and the stability properties of the resulting spatial structures are determined. Front location in the convective regime is highly sensitive to the upstream boundary condition and its dependence on this boundary condition is studied using a combination of numerical continuation and Monte Carlo simulations of the partial differential equation. Statistical properties of the system subjected to random or stochastic boundary conditions at the inlet are interpreted using the deterministic slow-fast spatial-dynamical system.

We improve the results by Siegelmann & Sontag (1995) by providing a novel and parsimonious constructive mapping between Turing Machines and Recurrent Artificial Neural Networks, based on recent developments of Nonlinear Dynamical Automata. The architecture of the resulting R-ANNs is simple and elegant, stemming from its transparent relation with the underlying NDAs. These characteristics yield promise for developments in machine learning methods and symbolic computation with continuous time dynamical systems. A framework is provided to directly program the R-ANNs from Turing Machine descriptions, in absence of network training. At the same time, the network can potentially be trained to perform algorithmic tasks, with exciting possibilities in the integration of approaches akin to Google DeepMind's Neural Turing Machines., Comment: 11 pages, 3 figures

The folded node is a singularity associated with loss of normal hyperbolicity in systems where mixtures of slow and fast timescales arise due to singular perturbations. Canards are special solutions that reveal a counteractive feature of the local dynamics, namely trajectories that flow from attractive regions of space into repulsive regions. An alternative way to model switches between timescales is using piecewise-smooth differential equations. There is presently no adequate theory or method for relating slow-fast and piecewise-smooth models. Here we derive the analogous piecewise-smooth system for the folded node by pinching phase space to sharpen the switch between timescales. The corresponding piecewise-smooth system contains a so-called two-fold singularity, and exhibits the same topology and number of canards as the slow-fast system. Thus pinching provides a piecewise-smooth approximation to a slow-fast system.

Short-Term Synaptic Plasticity (STSP) strongly affects the neural dynamics of cortical networks. The Tsodyks and Markram (TM) model for STSP accurately accounts for a wide range of physiological responses at different types of cortical synapses. Here, we report for the first time a route to chaotic behavior via a Shilnikov homoclinic bifurcation that dynamically organises some of the responses in the TM model. In particular, the presence of such a homoclinic bifurcation strongly affects the shape of the trajectories in the phase space and induces highly irregular transient dynamics; indeed, in the vicinity of the Shilnikov homoclinic bifurcation, the number of population spikes and their precise timing are unpredictable and highly sensitive to the initial conditions. Such an irregular deterministic dynamics has its counterpart in stochastic/network versions of the TM model: the existence of the Shilnikov homoclinic bifurcation generates complex and irregular spiking patterns and --acting as a sort of springboard-- facilitates transitions between the down-state and unstable periodic orbits. The interplay between the (deterministic) homoclinic bifurcation and stochastic effects may give rise to some of the complex dynamics observed in neural systems., Comment: Published in PNAS (Early edition) sept. 2013

We investigate the dynamics of large stochastic networks with different timescales and nonlinear mean-field interactions. After deriving the limit equations for a general class of network models, we apply our results to the celebrated Wilson-Cowan system with two populations with or without slow adaptation, paradigmatic example of nonlinear mean-field network. This system has the property that the dynamics of the mean of the solution exactly satisfies an ODE. This reduction allows to show that in the mean-field limit and in multiple populations with multiple timescales, noise induces canard explosions and Mixed-Mode Oscillations on the mean of the solution. This sheds new light on the qualitative effects of noise and sensitivity to precise noise values in large stochastic networks. We further investigate finite-sized networks and show that systematic differences with the mean-field limits arise in bistable regimes (where random switches between different attractors occur) or in mixed-mode oscillations, were the finite-size effects induce early jumps due to the sensitivity of the attractor.

We consider the chemostat model with the substrate concentration as the single measurement. We propose a control strategy that drives the system at a steady state maximizing the gas production without the knowledge of the specific growth rate. Our approach separates the extremum seeking problem from the feedback control problem such that each of the two subproblems can be solved with relatively simple algorithms. We are then free to choose any numerical optimization algorithm. We give a demonstration for two choices: one is based on slow-fast dynamics and numerical continuation, the other is a combination of golden-section and Newton iteration. The method copes with non-monotonic growth functions.

We propose an adaptive control law that allows one to identify unstable steady states of the open-loop system in the single-species chemostat model without the knowledge of the growth function. We then show how one can use this control law to trace out (reconstruct) the whole graph of the growth function. The process of tracing out the graph can be performed either continuously or step-wise. We present and compare both approaches. Even in the case of two species in competition, which is not directly accessible with our approach due to lack of controllability, feedback control improves identifiability of the non-dominant growth rate., Comment: expansion of ideas from proceedings paper (17 pages, 8 figures), proceedings paper is version v1

In multiple time-scale (singularly perturbed) dynamical systems, canards are counterintuitive solutions that evolve along both attracting and repelling invariant manifolds. In two dimensions, canards result in periodic oscillations whose amplitude and period grow in a highly nonlinear way: they are slowly varying with respect to a control parameter, except for an exponentially small range of values where they grow extremely rapidly. This sudden growth, called a canard explosion, has been encountered in many applications ranging from chemistry to neuronal dynamics, aerospace engineering and ecology. Canards were initially studied using nonstandard analysis, and later the same results were proved by standard techniques such as matched asymptotics, invariant manifold theory and parameter blow-up. More recently, canard-like behaviour has been linked to surfaces of discontinuity in piecewise-smooth dynamical systems. This paper provides a new perspective on the canard phenomenon by showing that the nonstandard analysis of canard explosions can be recast into the framework of piecewise-smooth dynamical systems. An exponential coordinate scaling is applied to a singularly perturbed system of ordinary differential equations. The scaling acts as a lens that resolves dynamics across all time-scales. The changes of local curvature that are responsible for canard explosions are then analysed. Regions where different time-scales dominate are separated by hypersurfaces, and these are pinched together to obtain a piecewise-smooth system, in which curvature changes manifest as discontinuity-induced bifurcations. The method is used to classify canards in arbitrary dimensions, and to derive the parameter values over which canards form either small cycles (canards without head) or large cycles (canards with head)., Comment: This paper has been withdrawn by the author as it is currently being reworked

In this work we study mixed mode oscillations in a model of secretion of GnRH (Gonadotropin Releasing Hormone). The model is a phantom burster consisting of two feedforward coupled FitzHugh-Nagumo systems, with three time scales. The forcing system (Regulator) evolves on the slowest scale and acts by moving the slow nullcline of the forced system (Secretor). There are three modes of dynamics: pulsatility (transient relaxation oscillation), surge (quasi steady state) and small oscillations related to the passage of the slow nullcline through a fold point of the fast nullcline. We derive a variety of reductions, taking advantage of the mentioned features of the system. We obtain two results; one on the local dynamics near the fold in the parameter regime corresponding to the presence of small oscillations and the other on the global dynamics, more specifically on the existence of an attracting limit cycle. Our local result is a rigorous characterization of small canards and sectors of rotation in the case of folded node with an additional time scale, a feature allowing for a clear geometric argument. The global result gives the existence of an attracting unique limit cycle, which, in some parameter regimes, remains attracting and unique even during passages through a canard explosion., Comment: 38 pages, 16 figures

The well-studied Hindmarsh-Rose model of neural action potential is revisited from the point of view of global bifurcation analysis. This slow-fast system of three paremeterised differential equations is arguably the simplest reduction of Hodgkin-Huxley models capable of exhibiting all qualitatively important distinct kinds of spiking and bursting behaviour. First, keeping the singular perturbation parameter fixed, a comprehensive two-parameter bifurcation diagram is computed by brute force. Of particular concern is the parameter regime where lobe-shaped regions of irregular bursting undergo a transition to stripe-shaped regions of periodic bursting. The boundary of each stripe represents a fold bifurcation that causes a smooth spike-adding transition where the number of spikes in each burst is increased by one. Next, numerical continuation studies reveal that the global structure is organised by various curves of homoclinic bifurcations. In particular the lobe to stripe transition is organised by a sequence of codimension-two orbit- and inclination-flip points that occur along {\em each} homoclinic branch. Each branch undergoes a sharp turning point and hence approximately has a double-cover of the same curve in parameter space. The sharp turn is explained in terms of the interaction between a two-dimensional unstable manifold and a one-dimensional slow manifold in the singular limit. Finally, a new local analysis is undertaken using approximate Poincar\'{e} maps to show that the turning point on each homoclinic branch in turn induces an inclination flip that gives birth to the fold curve that organises the spike-adding transition. Implications of this mechanism for explaining spike-adding behaviour in other excitable systems are discussed., Comment: 32 pages, 18 figures, submitted to SIAM Journal on Applied Dynamical Systems

Rapid action potential generation --- spiking --- and alternating intervals of spiking and quiescence --- bursting --- are two dynamic patterns observed in neuronal activity. In computational models of neuronal systems, the transition from spiking to bursting often exhibits complex bifurcation structure. One type of transition involves the torus canard, which was originally observed in a simple biophysical model of a Purkinje cell. In this article, we expand on that original result by showing that torus canards arise in a broad array of well-known computational neuronal models with three different classes of bursting dynamics: sub-Hopf/fold cycle bursting, circle/fold cycle bursting, and fold/fold cycle bursting. The essential features that these models share are multiple time scales leading naturally to decomposition into slow and fast systems, a saddle-node of periodic orbits in the fast system, and a torus bifurcation in the full system. We show that the transition from spiking to bursting in each model system is given by an explosion of torus canards. Based on these examples, as well as on emerging theory, we propose that torus canards are a common dynamic phenomenon separating the regimes of spiking and bursting activity.