Cooperation in neural systems: Bridging complexity and periodicity

Abstract

Inverse power law distributions are generally interpreted as a manifestation of complexity, and waiting time distributions with power index μ < 2 reflect the occurrence of ergodicity-breaking renewal events. In this paper we show how to combine these properties with the apparently foreign clocklike nature of biological processes. We use a two-dimensional regular network of leaky integrate-and-fire neurons, each of which is linked to its four nearest neighbors, to show that both complexity and periodicity are generated by locality breakdown: Links of increasing strength have the effect of turning local interactions into long-range interactions, thereby generating time complexity followed by time periodicity. Increasing the density of neuron firings reduces the influence of periodicity, thus creating a cooperation-induced renewal condition that is distinctly non-Poissonian.

Complexity matching in neural networks

“…We find that the adoption of avalanche size as indicator of criticality leads to a value that corresponds to the supercritical regime, a result confirming the observation of the earlier work of [20]. In this supercritical condition the renewal character of neuron firings is lost, and a bridge between temporal complexity and periodicity is established [10]. This result casts doubt on the equivalence between temporal complexity and self organized criticality (SOC).…”

“…As discussed in the earlier work of [10,19,20], at criticality, the time distance between two consecutive events fits the prescription of equation (1), with the survival probabilityΨ t ( ) corresponding to the waiting time distribution density ψ t ( ) of equation (1) taking the specific shape of a Mittag-Leffler (ML) function (see section 4 for more information about this form of survival probability). The events have been proven to be renewal by using the aging experiment method [10,20]. In this article we take the renewal condition for granted and we assume that the emergence of ML survival probability is a sign of temporal complexity.…”

“…The same aging experiment method was used to prove that a set of cooperating dipoles at criticality generate renewal fluctuations [9]. The aging experiment method was recently adopted to study cooperation in neural systems [10]. It was used to prove that cooperation between neurons generates a phase transition to a process that, for large values of the cooperation parameter, becomes periodic after crossing an intermediate nonrenewal regime, and to confirm that at criticality the process is renewal.…”

Renewal and memory origin of anomalous diffusion: A discussion of their joint action

“…Another area of application is in the analysis of brain dynamics [20] adopting the hypothesis that the brain undergoes phase transition. This theoretical perspective leads to the conjecture that criticality generates renewal events and with them 1/f noise, without ruling out the possibility of a joint action of renewal and memory [21]. The single column of a surface growing, thanks to the experimental procedure of molecular epitaxy [18], under specific conditions of cooperation with the other units, is expected to show signs of criticality and with it renewal events properly described by the Mittag-Leffler function.…”

Effect of noise and detector sensitivity on a dynamical process: Inverse power law and Mittag-Leffler interevent time survival probabilities

“…There are numerical results on neural dynamics showing that the ML process may be generated by cooperation of limited intensity [19,30]. The results of this paper may be used to establish for which value of the cooperation parameter the distribution of the time distances between two consecutive neuron firings makes a transition from the exponential to the stretched exponential form.…”