Switching in Cerebellar Stellate Cell Excitability in Response to a Pair of Inhibitory/Excitatory Presynaptic Inputs: A Dynamical System Perspective

Abstract

Cerebellar stellate cells form inhibitory synapses with Purkinje cells, the sole output of the cerebellum. Upon stimulation by a pair of varying inhibitory and fixed excitatory presynaptic inputs, these cells do not respond to excitation (i.e., do not generate an action potential) when the magnitude of the inhibition is within a given range, but they do respond outside this range. We previously used a revised Hodgkin–Huxley type of model to study the nonmonotonic first-spike latency of these cells and their temporal increase in excitability in whole cell configuration (termed run-up). Here, we recompute these latency profiles using the same model by adapting an efficient computational technique, the two-point boundary value problem, that is combined with the continuation method. We then extend the study to investigate how switching in responsiveness, upon stimulation with presynaptic inputs, manifests itself in the context of run-up. A three-dimensional reduced model is initially derived from the original six-dimensional model and then analyzed to demonstrate that both models exhibit type 1 excitability possessing a saddle-node on an invariant cycle (SNIC) bifurcation when varying the amplitude of [Formula: see text]. Using slow-fast analysis, we show that the original model possesses three equilibria lying at the intersection of the critical manifold of the fast subsystem and the nullcline of the slow variable [Formula: see text] (the inactivation of the A-type K[Formula: see text] channel), the middle equilibrium is of saddle type with two-dimensional stable manifold (computed from the reduced model) acting as a boundary between the responsive and non-responsive regimes, and the (ghost of) SNIC is formed when the [Formula: see text]-nullcline is (nearly) tangential to the critical manifold. We also show that the slow dynamics associated with (the ghost of) the SNIC and the lower stable branch of the critical manifold are responsible for generating the nonmonotonic first-spike latency. These results thus provide important insight into the complex dynamics of stellate cells.

Bursting in cerebellar stellate cells induced by pharmacological agents: Non-sequential spike adding

“…, where x i 's are the state variables of the full system (1). We adapt a two-point boundary value problem (2PBVP) technique to compute the profiles and stable manifolds as described in [17]. The codes for regenerating the figures are available online [48].…”

“…As suggested above, the activation of I Na (m 1 ) and I T (m T,1 ) are assumed to be fast and set to steady state. The full model, given by (1) and (2), has been reparametrized in a manner similar to the method used in [16,17] to capture the profile of the action potential (AP)-cycle in the ðV; _ V Þ-plane. The resulting parameters are kept fixed throughout the paper, unless otherwise stated.…”

“…The resulting parameters are kept fixed throughout the paper, unless otherwise stated. A full list of these parameters is provided in Tables 1 and 2 under control condition. Time-scale separation of the full system (1) can be performed in a manner similar to that outlined in [17] (see Fig 1 in that paper). This is done by first plotting the time courses of the derivatives of its state variables and then classifying them as fast/slow variables according to how much variation they exhibit in their time courses: the larger the variation the faster the variable.…”

“…It was shown in [13,16,17] that the revised Hodgkin-Huxley model, comprised of Na + , K + , leak, A-type K + and T-type Ca 2+ currents, was able to produce the four key features of CSCs, including, type I excitability, the non-monotonic first-spike latency, runup and switching in responsiveness. Indeed, using 2PBVP, the continuation method in AUTO and slow-fast analysis, the dynamics underpinning these activities were elucidated.…”

“…We have previously analyzed the main features of CSCs, including type I excitability, runup (i.e., temporal increase in excitability) non-monotonic first-spike latency and switching in responsiveness when a pair of inhibitory and excitatory post-synaptic inputs are applied [16,17]; a revised Hodgkin-Huxley type model was used to accomplish this. The model was originally developed in [18], but later reparametrized to capture more accurately the cycle of tonic firing in ðV; _ V Þ-plane during both pre-and post-runup conditions [13].…”