Repetitive Action Potential Firing

Abstract

The biophysical basis of repetitive action potential firing can be understood in terms of the opposing positive and negative feedback processes that generate the repetitive activity. The relative timescales of these processes, which can include ion channel activation and inactivation as well as ion accumulation within a cell, are key factors in the generation of repetitive single‐spike firing, intrinsic burst firing and repetitive activity resulting from network interactions. Bifurcation theory, in concert with the steady state current–voltage relationship, explains why the onset of repetitive firing can be gradual or abrupt and accounts for different excitability types and distinct types of bistability. Moreover, bifurcation theory allows for the classification of different mechanisms underlying rhythmic bursting. Key Concepts A branch of mathematics called bifurcation theory can explain initiation of repetitive firing without reference to the specific ionic currents involved. Two major bifurcations can underlie the initiation of repetitive firing. Initiation of repetitive firing is gradual if the underlying bifurcation is a saddle‐node on an invariant circle (SNIC). A SNIC requires that the steady state IV curve intersect the zero current axis in three places. Initiation of repetitive firing is abrupt at a threshold frequency if due to a Hopf bifurcation. Most rhythmic bursting can be classified as square wave, parabolic, or elliptical, depending upon the bifurcations that initiate and terminate spiking. Rhythmic bursting requires at least one slow process, and parabolic bursting requires two. Repetitive firing can also arise via network mechanisms, in addition to the intrinsic ones described above.

Inactivation mode of sodium channels defines the different maximal firing rates of conventional versus atypical midbrain dopamine neurons

“…If the AHP meets this condition, then pacemaking can be arbitrarily slow and the mathematical term for the bifurcation is a saddle node on an invariant limit cycle. If not, then the stable rest potential is bistable with pacemaking for some range of applied current, there is an abrupt onset of pacemaking at a minimum threshold frequency, and the mathematical name for the bifurcation is a saddle node not on an invariant limit cycle [ 52 , 53 ]. The shallower AHP of the atypical population makes them more likely to exhibit a saddle node not on an invariant limit cycle bifurcation ( S1A1 Fig ) at the onset of spiking compared to the conventional ( S1B1 Fig ) which are more likely to exhibit a saddle node bifurcation in an invariant limit cycle).…”

“…2008 blocked AMPA/ NMDA and GABA-A synaptic receptors with 20 μM CNQX and 10 μM gabazine respectively, whereas in the present study synaptic channels were blocked with 12.5 μM CNQX, 4 μM gabazine, and 10 μM of the NMDA specific blocker DL-AP5. [52,53], if the AHP is too shallow for the action potential to reach or overshoot the saddle node. Consistent with failure of spiking at small amplitude at large DC, ramp currents, oscillations terminate via a super critical Hopf (SupH).…”

“…Bifurcation diagram for atypical model repeated from Fig 4D . Type 2 firing in the atypical model can potentially result from a saddle node (SN) bifurcation not on an invariant circle [ 52 , 53 ], if the AHP is too shallow for the action potential to reach or overshoot the saddle node. Consistent with failure of spiking at small amplitude at large DC, ramp currents, oscillations terminate via a super critical Hopf (SupH).…”

Polynomial, piecewise-Linear, Step (PLS): A Simple, Scalable, and Efficient Framework for Modeling Neurons

“…Note that any continuous phenomenological model, such as FitzHugh-Nagumo (FitzHugh, 1955 , 1961 ) or Morris-Lecar (Morris and Lecar, 1981 ) can reproduce plateau-potentials, just because of the continuity of the dynamical system. On the other hand, models with resetting can reproduce phenomena which are not possible to model if the system is continuous and low-dimensional; for example, a model the intrinsic bursting cell-type requires a 3D continuous system (Hindmarsh and Rose, 1984 ; Drion et al, 2015 ; Knowlton et al, 2020 ), but can be reproduced in 2D discontinues models (Izhikevich, 2003 , 2004 ; Brette and Gerstner, 2005 ; Destexhe, 2009 ). The hybrid nature of discontinuous models, which mix continuous dynamics with mapping [resetting (Izhikevich, 2010 )], reduces the number of differential equations and the required elementary operations per second of model time.…”

“…Saddle-node bifurcation is necessary but not sufficient to produce arbitrary slow oscillations because type 1 excitability needs a trajectory that passes through the half-stable fixed point at the bifurcation (Knowlton et al, 2020 ). In other words, a limit circle should go through the saddle-node, and therefore this bifurcation is known as a saddle-node on an invariant curve or cycle (SNIC).…”