Canards of mixed type in a neural burster.

Canards are solutions of slow-fast systems that spend long times near branches of repelling equilibria, periodic orbits, or higher-dimensional invariant sets. Here, we report on the observation of a new type of canard orbit, labeled a canard of mixed type. This canard orbit is a hybrid of the classical limit cycle canards, which spend long times near attracting and repelling branches of equilibria, and torus canards, which spend long times near attracting and repelling branches of periodic orbits. The canards of mixed type arise in a model of neural bursting activity of fold-fold cycle type, and, as other canard phenomena, separate different dynamic states.

Alpha-frequency rhythms desynchronize over long cortical distances: a modeling study.

Neocortical networks of excitatory and inhibitory neurons can display alpha(a)-frequency rhythms when an animal is in a resting or unfocused state. Unlike some gamma- and beta-frequency rhythms, experimental observations in cats have shown that these alpha-frequency rhythms need not synchronize over long cortical distances. Here, we develop a network model of synaptically coupled excitatory and inhibitory cells to study this asynchrony. The cells of the local circuit are modeled on the neurons found in layer V of the neocortex where alpha-frequency rhythms are thought to originate. Cortical distance is represented by a pair of local circuits coupled with a delay in synaptic propagation. Mathematical analysis of this model reveals that the h and T currents present in layer V pyramidal (excitatory) cells not only produce and regulate the alpha-frequency rhythm but also lead to the occurrence of spatial asynchrony. In particular, these inward currents cause excitation and inhibition to have nonintuitive effects in the network, with excitation delaying and inhibition advancing the firing time of cells; these reversed effects create the asynchrony. Moreover, increased excitatory to excitatory connections can lead to further desynchronization. However, the local rhythms have the property that, in the absence of excitatory to excitatory connections, if the participating cells are brought close to synchrony (for example, by common input), they will remain close to synchrony for a substantial time.