Pulse coupled oscillators and the phase resetting curve

Abstract

Limit cycle oscillators that are coupled in a pulsatile manner are referred to as pulse coupled oscillators. In these oscillators, the interactions take the form of brief pulses such that the effect of one input dies out before the next is received. A phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike in an oscillatory neuron depending upon where in the cycle the input is applied. PRCs can be used to predict phase locking in networks of pulse coupled oscillators. In some studies of pulse coupled oscillators, a specific form is assumed for the interactions between oscillators, but a more general approach is to formulate the problem assuming a PRC that is generated using a perturbation that approximates the input received in the real biological network. In general, this approach requires that circuit architecture and a specific firing pattern be assumed. This allows the construction of discrete maps from one event to the next. The fixed points of these maps correspond to periodic firing modes and are easier to locate and analyze for stability compared to locating and analyzing periodic modes in the original network directly. Alternatively, maps based on the PRC have been constructed that do not presuppose a firing order. Specific circuits that have been analyzed under the assumption of pulsatile coupling include one to one lockings in a periodically forced oscillator or an oscillator forced at a fixed delay after a threshold event, two bidirectionally coupled oscillators with and without delays, a unidirectional N-ring of oscillators, and N all-to-all networks.

Multistable Jittering in Oscillators with Pulsatile Delayed Feedback

“…1(a)] the emitted pulses affect the oscillator after a delay τ at the time instant t * = t+τ . When the pulse is received, the phase of the oscillator undergoes an instantaneous shift by an amount ∆ϕ = Z(ϕ(t * − 0)), where Z(ϕ) is the phase resetting curve (PRC).Thus, the dynamics of the oscillator can be described by the following equation [3,[10][11][12][13]:where t j are the instants when the pulses are emitted. Note that we adopt the convention Z ex (ϕ) := 0.1 sin q (πϕ) , where q controls the steepness of Z ex (ϕ) [see Fig.…”

“…Interaction via pulse-like signals is important in neuron populations [1][2][3], biological [4,5], optical and optoelectronic systems [6]. Often, time delays are inevitable in such systems as a consequence of the finite speed of pulse propagation [7].…”

Single Synapse Information Coding in Intraburst Spike Patterns of Central Pattern Generator Motor Neurons

“…For instance, phase response curves (PRCs) (Ayers and Selverston, 1984;Canavier and Achuthan, 2010) have greatly contributed to our understanding of synchronization in CPGs, considering neurons as periodic oscillators (Oprisan et al, 2004;Galán et al, 2005;Ermentrout et al, 2007;Schleimer and Stemmler, 2009). PRC methods target the phase shifts produced in periodic bursts by interacting neurons, disregarding any oscillations of intraburst ISIs.…”

“…2 A). The same choice is often used in other techniques applied to pyloric neurons time series analysis (Ayers and Selverston, 1984;Canavier and Achuthan, 2010). The PD neuron is part of the pacemaker group, which strongly inhibits all other motor neurons, dictating the pyloric rhythm.…”

Computação por assembleias neurais em redes neurais pulsadas.

“…Geralmente, o método mais usado para analisar e rastrear quanto um pulso de entrada avança ou atrasa o próximo pulso é a curva de resposta de fase (PRC -phase response curve) (análise de diferentes redes PCO em [114] Os autores apontam similaridades entre os circuitos corticais e os circuitos de CPGs na anatomia, na biofísica, nas patologias, na modularidade, na dinâmica; embora a principal diferença esteja na plasticidade dos circuitos: circuitos do córtex são muito mais plásticos. Em outras palavras, "[o córtex pode ser formado por] um tipo especial de CPG baseado em assembleias hebbianas específico para aprendizado e armazenagem ou recuperação de memórias (um "CPG aprendiz" ou "CPG memória")"…”