Effects of conduction delays on the existence and stability of one to one phase locking between two pulse-coupled oscillators

Abstract

Gamma oscillations can synchronize with near zero phase lag over multiple cortical regions and between hemispheres, and between two distal sites in hippocampal slices. How synchronization can take place over long distances in a stable manner is considered an open question. The phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike, depending upon where in the cycle it is received. We use PRCs under the assumption of pulsatile coupling to derive existence and stability criteria for 1:1 phase-locking that arises via bidirectional pulse coupling of two limit cycle oscillators with a conduction delay of any duration for any 1:1 firing pattern. The coupling can be strong as long as the effect of one input dissipates before the next input is received. We show the form that the generic synchronous and anti-phase solutions take in a system of two identical, identically pulse-coupled oscillators with identical delays. The stability criterion has a simple form that depends only on the slopes of the PRCs at the phases at which inputs are received and on the number of cycles required to complete the delayed feedback loop. The number of cycles required to complete the delayed feedback loop depends upon both the value of the delay and the firing pattern. We successfully tested the predictions of our methods on networks of model neurons. The criteria can easily be extended to include the effect of an input on the cycle after the one in which it is received.

Globally attracting synchrony in a network of oscillators with all-to-all inhibitory pulse coupling

“…Synchrony with $\delta =0$ is always unstable because according to [20] and [37] perturbing the synchrony by slightly perturbing the firing pattern of half the neurons synchronous cluster to yields the eigenvalue $\lambda =false(1-f_{2,0}false(0^{+}false)false)false(1-f_{2,0}false(1^{-}false)false)$ or $\lambda =false(1-f_{2,0}false(0^{+}false)false)false(1-f_{2,0}false(1^{-}false)false)$, either of which results in instability due to an eigenvalue with an effectively infinite absolute value as a phase of one is approached from the left. In a synchronous mode with $\delta >0$, for small $\delta >0$ each cluster receives the input from the other cluster at a phase of $\frac{\delta }{P_{n,\delta }}={\phi }_S$, and treating two clusters as two oscillators the delay switches the form of the eigenvalue to $\lambda =1-2f_{2,0}false(\frac{\delta }{P_{n,\delta }}false)$ [20]. Thus a PRC slope at the input phase of $0<f^{\prime }false({\phi }_{S}false)<1$ guarantees stability of global synchrony, provided the stimulus interval is less than the network period, that is, for $\delta <P_{n,\delta }$.…”

“…In brief, the effect of a pulse in one oscillator will take more than one cycle to affect the timing of a spike in the same oscillator for longer delays. However, firing patterns in which a spike in one oscillator takes more than one cycle to affect the firing of another spike in the same oscillator via feedback through the network are in general less stable than those in which feedback is received within a single cycle [20], so in practice the magnitude of the delay should not have an upper limit. In any case, we are interested in cases in which the two cluster mode does not exist.…”

“…Systems with conduction delays [18,19] are of particular interest in neurobiology and other applications such as laser arrays, electronic circuits, microwave devices and communications satellites. In some cases, conduction delays can stabilize synchrony [20–22]. We use phase resetting theory [23–25] and stability results based on event driven maps to prove that in a network of pulse coupled phase oscillators with a small conduction delay $\delta$, inhibition can be globally synchronizing.…”

“…Some results have been obtained for sparse coupling: for inhibitory pulse coupling with arbitrary coupling, Timme and Wolf [29] showed that synchrony is locally stable provided the synaptic conductances are normalized so that each neuron receives the same amount of total inhibitory conductance, that the implicitly defined phase resetting curve has the appropriate stabilizing shape, and that there is a path through the network that connects every oscillator to every other oscillator. The derivation also included a small delay, and, although it is not noted in the study of Timme and Wolf, this small delay is required [20,21,30] in order to avoid the highly destabilizing discontinuity as a phase of one is approached from the left (Figure 1, solid trace), as proven in this study for networks of N neurons.…”

Direct connections assist neurons to detect correlation in small amplitude noises

“…In the absence of noise, any finite value of the forward coupling strength can lead to a zone of 1:1 synchrony, in which the dissimilar neurons fire in a causal master-slave fashion (Takahashi et al, 2009; Bayati and Valizadeh, 2012). In such causal limit the postsynaptic neuron fires immediately after receiving presynaptic stimulation (Woodman and Canavier, 2011; Wang et al, 2012). In our model delays in communication have been ignored, so in the causal 1:1 synchrony zones the postsynaptic neuron fires just one simulation time step after the firing of presynaptic neuron.…”