Modeling positive Granger causality and negative phase lag between cortical areas

Abstract

Different measures of directional influence have been employed to infer effective connectivity in the brain. When the connectivity between two regions is such that one them (the sender) strongly influences the other (the receiver), a positive phase lag is often expected. The assumption is that the time difference implicit in the relative phase reflects the transmission time of neuronal activity. However, Brovelli et al. (2004) observed that, in monkeys engaged in processing a cognitive task, a dominant directional influence from one area of sensorimotor cortex to another may be accompanied by either a negative or a positive time delay. Here we present a model of two brain regions, coupled with a well-defined directional influence, that displays similar features to those observed in the experimental data. This model is inspired by the theoretical framework of Anticipated Synchronization developed in the field of dynamical systems. Anticipated Synchronization is a form of synchronization that occurs when a unidirectional influence is transmitted from a sender to a receiver, but the receiver leads the sender in time. This counterintuitive synchronization regime can be a stable solution of two dynamical systems coupled in a masterslave (sender-receiver) configuration when the slave receives a negative delayed self-feedback. Despite efforts to understand the dynamics of Anticipated Synchronization, experimental evidence for it in the brain has been lacking. By reproducing experimental delay times and coherence spectra, our results provide a theoretical basis for the underlying mechanisms of the observed dynamics, and suggest that the primate cortex could operate in a regime of Anticipated Synchronization as part of normal neurocognitive function.

Anticipating chaotic synchronization via act-and-wait coupling

“…It has been also observed in excitable systems driven by noise [5,13]. Recently, an anticipating spike synchronization has been detected in two unidirectionally coupled nonidentical neurons without the time-delay self-feedback in a slave system [14] as well as in unidirectionally coupled networks of neurons [15,16].…”

“…The evolution matrix (t) of this system can be obtained analytically, and the Floquet multipliers μ 1,2 are defined as the eigenvalues of the 2 × 2 monodromy matrix (2τ ), det[ (2τ ) − μI] = 0. The origin of the system (16) is stable if the both Floquet multipliers are inside the unit circle of the complex plane. Analytical expressions for these multipliers are rather complex and we do not present them here; instead, we show graphically the dependence of the Floquet exponents λ 1,2 = ln(μ 1,2 )/2τ on the coupling strength in Fig.…”

Signal prediction by anticipatory relaxation dynamics

“…In particular, it could be worthwhile to search for signatures of ARD in, for example, the distribution of peak prediction variability in cognitive motor control experiments [48]. Recently anticipatory dynamics in brain signaling have been observed and explained by time delays mediated by synaptic coupling properties [49]. Interestingly, the relationship τ C < τ recently has been observed in a manual tracking experiment of chaotic signals with external feedback delay [50], a task whose execution most likely does not involve an internal model of the signal system in untrained subjects.…”

Anticipated synchronization in neuronal circuits unveiled by a phase-response-curve analysis

“…Up to now, we focussed on the DS/AS transition assuming identical free-running periods for all the neurons [9,11,13]. The results presented in the previous section also assumed all periods to be identical.…”

“…Using the Hodgkin-Huxley model, a transition from the more intuitive delayed-synchronization (DS) regime to the AS regime was found when changing the inhibitory conductance impinging on the receiver neuron. Recently, the ideas introduced in [9] were extended to neuronal populations [11] to explain the observations of a positive Granger Causality, with well-defined directional influence, accompanied by either a positive or negative phase lag in the recordings of the motor cortex activity of monkeys while doing a visual task [18,19]. Despite the interest attracted by AS in neuronal circuits, * fernanda@fis.ufal.br a thorough understanding of how this particular state is stablished is missing.…”