CS14: Complex systems II

date: 7/14/2025, time: 16:00-17:30, room: ICS 118

Organizer: Marcin Magdziarz (Wrocław University of Science and Technology)

Chair: Krzysztof Burnecki (Wrocław University of Science and Technology)

Arcsine laws for Brownian motion with resetting

Marcin Magdziarz (Wroclaw University of Science and Technology)

Abstract: The classical arcsine laws describe surprising statistical properties of Brownian motion, capturing the distribution of specific random variables associated with its temporal evolution. In this work, we explore analogous phenomena for Brownian motion under Poissonian resetting, a process that introduces interruptions at random times, fundamentally altering its dynamics. We derive closed-form expressions for the probability density functions corresponding to the first and second arcsine laws, highlighting their distinct behavior under resetting. Additionally, we present numerical insights into the third arcsine law, providing a comprehensive perspective on this generalization of classical results.

Bibliography

\([1]\) Kacper Taźbierski, Marcin Magdziarz. "Arcsine laws for Brownian motion with Poissonian resetting." Chaos, vol. 35, 2025, pp. 023163.

Dynamics, bifurcations and randomness in integrate and fire neuron models

Piotr Kowalczyk (Wrocław University of Science and Technlogy)

We present the investigations of a class of minimal neuron models which are termed “integrate and fire models”\([1,2]\). These models capture so-called spiking and bursting behaviour of neurons \([2]\). We focus on phase space transitions which lead to spike increment, that is an increased number of voltage spikes exhibited by neurons which occur in our model systems. In particular, we focus on a specific type of phase space transition (bifurcation) which is triggered by the presence of a limit cycle, which we term a canard cycle, and resets. We explain the mechanism behind this transition \([4]\), and apply our analytical results to a class of conductance based adaptive integrate and fire models \([3,5]\), which through increased complexity capture some biological aspects of neuronal conduction. Lastly, we present the results of a numerical experiment aimed at investigating the effects of noise on phase space transitions explained analytically.

Consider the following family of adaptive integrate and fire models \[\tag{1} v' = |v|-w+I\\ w' = \varepsilon F(v,w), %\label{eq:modelgen}\] where \(I\) is a regular parameter representing an applied constant current, \(0<\varepsilon\ll 1\) is a small parameter, with the limit \(\varepsilon \rightarrow 0\) relevant to the full system, and \(F\) is a smooth function, which we will take to be linear in both arguments for simplicity, and without loss of generality on the dynamics of interest. We append to system \((1)\) the following reset rule: for \(v=v_{\mathrm{thr}}\), we have \[\tag{2}%\label{eq:reset} (v, \;w) \longrightarrow \left(v_{\mathrm{res}},\; w+k\right),\] meaning that as soon as \(v\) reaches some threshold value \(v_{\mathrm{thr}}\), the values of state variables \(v\) and \(w\) are reset to \(v_{\mathrm{res}}\) and \(w+k\), respectively. System \((1)\) is parametrised by the fast time \(\tau\) and it is useful to rescale time so as to introduce the slow time \(t=\varepsilon\tau\), which brings system \((1)\) in its alternative slow-time formulation, namely \[\tag{3} \varepsilon\dot{v} = |v|-w+I\\ \dot{w} = F(v,w). %\label{eq:modelgenbis}\] The reset rule \((2)\) remains unchanged in the alternative slow-time formulation \((3)\).

The main results are summarised by the following two propositions and a theorem:

Proposition 1. Consider system \((1)\)-\((2)\) with positive parameters \(v_{\mathrm{thr}} = {\mathcal{O}}(1)\), \(I = \mathcal{O}(v_{\mathrm{res}})\). Then for every \(\varepsilon>0\) small enough and \(w_0 > (1+\varepsilon)(v_{\mathrm{res}} + I)\), there exist some \(k > 0\) and \(N \geq 1\) such that the system possesses a limit cycle characterised by \(N\geq 1\) resets, and a periodic point \((v_{\mathrm{res}}, w_0)\). Theorem 1. A major result. Consider system \((1)\)-\((2)\) with positive parameters \(v_{\mathrm{thr}} = {\mathcal{O}}(1) \gg v_{\mathrm{res}}\), \(I = {\mathcal{O}}(v_{\mathrm{res}})\). Then for every \(\varepsilon>0\) small enough and \(w_0 = (1+\varepsilon)(v_{\mathrm{res}} + I)\), there exist \(k > 0\) and \(N\geq 1\) such that the system possesses an \(N\)-reset periodic cycle and a periodic point \((v_{\mathrm{res}}, w_0)\) with the following set of inequalities: \[\tag{4} w_0 = (1+\varepsilon)(v_{\mathrm{res}} + I) = w_{N} > w_{N-1} > w_{N-2} >\ldots > w_1, %\label{eq:exstCN}\] where \(w_i\), \(1\leq i\leq N-1\), are the values of the \(w\)-component of the system immediately after the \(i\)th reset. Such a periodic cycle is a canard cycle.

Proposition 2. Consequences of Propositions 1 and Theorem 1. Consider system \((1)\)-\((2)\) with parameters \(v_{\mathrm{thr}} = {\mathcal{O}}(1) \gg v_{\mathrm{res}}\), \(I = {\mathcal{O}}(v_{\mathrm{res}})\), and some \(k^*\), all positive. Let us further assume that a canard cycle with \(N\) number of resets as described by Theorem 1 exists for a given set of parameter values. Then for some \(k\), sufficiently close to but different from \(k^*\), there exists a periodic cycle in the system characterised by \(N + 1\) of resets, and a periodic point \((v_{\mathrm{res}}, w_0)\) such that the inequality \[w_0 = (1+\varepsilon)(v_{\mathrm{res}} + I) = w_{N + 1} > w_{N} > w_{N-1} > w_{N-2} >\ldots > w_1 \label{eq:exstCNp1}\] holds. Moreover, we have that \(|w_{N+1}-w_N| \ll \varepsilon\). Such a periodic cycle is a canard cycle.

The essential conclusion from the above propositions and the theorem is that the spike increment scenario is a discontinuous event in the parameter space, howevever the discontinuity is of the size smaller then some \(\varepsilon\), and for spike adding to take place a so-called Canard cycle, that is a limit cycle with a segment of the trajectory along the repelling part of the slow manifold, has to be present in the system. We will show numerical evidence that this scenario is typical to a large class of integrate and fire models, see \([5]\). Finally, the effect of noise on the spike adding scenario discussed here will be investigated numerically.

Bibliography

\([1]\) Izhikevich, E. M. "Simple model of spiking neurons". IEEE Transactions on neural networks, vol. 14, no. 6, Year 2003, pp. 1569–1572.

\([2]\) Coombes, S., Thul, R. and Wedgwood, K.C.A. "Nonsmooth dynamics in spiking neuron models". Physica D, vol. 241, Year 2012, pp. 2042-2057.

\([3]\) Górski, T., Depannemaecker, D. and Destexhe, A. "Conductance-based adaptive exponential integrate-and-fire model". Neural Computation, vol. 33, no. 1, Year 2021, pp. 41–66.

\([4]\) Desroches, M., Kowalczyk, P. and Rodrigues, S. "Spike-adding and reset-induced canard cycles in adaptive integrate and fire models" Nonlinear Dynamics, vol. 104, no. 3, Year 2021, pp. 2451–2470.

\([5]\) Desroches, M., Kowalczyk, P. and Rodrigues, S. "Discontinuity induced dynamics in Conductance-Based Adaptive Exponential Integrate-and-Fire Model" Bulletin of Mathematical Biology, vol. 87, no. 2, Year 2025.

The role of the fractional material derivative in Lévy walks

Marek Teuerle (Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology)

The Lévy walk model, a key framework for describing anomalous diffusion processes with finite-second moments, is distinguished by its intrinsic coupling of jump lengths and waiting times \([1]\). This model has proven invaluable for capturing the dynamics of diverse systems, from the collective movement of biological organisms like bacteria and marine predators to physical phenomena involving cold atoms and blinking quantum dots (see \([2,3]\) for examples).

Recent theoretical advances have clarified the macroscopic behavior of Lévy walks. It has been shown that their scaling limits, under Skorokhod’s \(\mathbb{J}_1\) convergence, are \(\alpha\)-stable processes governed by a strongly dependent inverse \(\alpha\)-stable subordinator or by some combinations of these kind of processes \([4]\). It is also a well-established fact that the governing dynamics of the Lévy walks’ scaling limit can be described by the fractional material derivative, an extension of the classical material derivative to fractional calculus \([4,5]\).

This presentation focuses on the numerical approximation of this fractional material derivative. We introduce a novel finite-volume upwind scheme that incorporates spatiotemporal coupling of the underlying process and builds upon known techniques for fractional derivatives \([6,7]\). We confirm the stability of our proposed method. Furthermore, we will demonstrate the scheme’s accuracy by applying it to a one-sided probability problem (jumps/displacements only to the left or only to the right) associated with Lévy walks and comparing the results with traditional Monte Carlo simulations. Finally, we will outline an extension of our numerical approach to accommodate Lévy walks with arbitrary step directions, so-called biased Lévy walks.

Bibliography

\([1]\) J. Klafter, A. Blumen, M.F. Shlesinger. "Random walks with infinite spatial and temporal moments." J. Stat. Phys., vol.27, 1982, pp. 499–512.

\([2]\) R. Metzler, J. Klafter. "The random walk’s guide to anomalous diffusion: a fractional dynamics approach." Phys. Rep., vol. 339, 2000, pp. 1–77.

\([3]\) V. Zaburdaev, S. Denisov, J. Klafter. "Lévy walks." Rev. Mod. Phys., vol. 87, 2015, pp. 483–530.

\([4]\) M. Magdziarz, M. Teuerle. "Asymptotic properties and numerical simulation of multidimensional Lévy walks." Commun. Nonlinear Sci. Numer. Simul., vol. 20, 2015, pp. 489–505.

\([5]\) I.M. Sokolov, R. Metzler. "Towards deterministic equations for Lévy walks: The fractional material derivative." Phys. Rev. E., vol. 67, 2003, pp. 010101(R).

\([6]\) Ł. Płociniczak. "Linear Galerkin-Legendre spectral scheme for a degenerate nonlinear and nonlocal parabolic equation arising in climatology." Appl. Numer. Math., vol. 179, 2022, pp. 105–124.

\([7]\) M. Teuerle, Ł. Płociniczak. "From Lévy walks to fractional material derivative: Pointwise representation and a numerical scheme." Commun. Nonlinear Sci. Numer. Simul., vol. 139, 2024, pp. 108316.