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The analysis above suggests that clustered firing patterns may arise due to noise perturbations to a periodic bursting regime. In order to further understand these dynamics, a fast-slow analysis was performed on the deterministic system within this regime. We chose parameters to be gAHP= 0.425and gh= 2.8, which results in periodic bursts of three action potentials. We first examined simulations, which revealed two variables operating with a slow time scale, namely mNaP and hKas (Figure 2.10A). Keeping the two slow variables fixed, the remaining (fast) subsys- tem was subjected to a numerical bifurcation analysis, which revealed two bifur- cations of importance for describing the bursting dynamics (see Figure 2.10B). For low values of mNaP, there exists a stable steady state which loses stability via a subcritical Hopf bifurcation (denoted SCH1) as mNaP is increased (marked by a dashed red line inFigure 2.10B). For high values of mNaP there exists a stable periodic orbit of period 1, which disappears via a homoclinic bifurcation (denoted HC1 and marked by a dotted red line inFigure 2.10B) as mNaP is decreased. Be- tween these two bifurcations there is a region of bistability between the steady state and the periodic orbit. These bifurcations in mNaP are drawn over a range of values of hKasinFigure 2.10B. A full bifurcation diagram and example bistable region for mNaP for hKas= 0.19is shown inFigure 2.11.

Plotting the periodic solution of the full subsystem in the two variables (mNaP and hKas, Figure 2.10B) is sufficient to describe the bursting dynamics. The tra- jectory follows a hysteresis loop through the fast subsystem. Beginning in the quiescent period between bursts, the two slow variables will be at a position in phase space such that the fast subsystem is on the steady state branch. The pe- riodic solution’s trajectory then moves along the steady state branch until SCH1 is reached, at which point the fast subsystem moves to the periodic orbit branch. This initiates the burst, with action potentials firing while slow variables move along the periodic orbit branch towards HC1. Once HC1 is reached, the burst

Results -50 0 50 V [mV] 0.585 0.6 0.615 m NaP 0 _{Time [s]} 5 0.182 0.19 0.198 h Kas 0.59 0.6 0.61 m NaP 0.18 0.185 0.19 0.195 0.2 h Kas -1 0 1 2 3 4 m slow -60 -40 -20 0 20 40 V [mV] 0.59 0.6 0.61 m slow -60 -40 -20 0 20 40 V [mV] C D A B HC2 HC2 HC1 SCH1 SCH2 SCH2 SCH3 SNP1

Figure 2.10: Fast-slow analysis of deterministic bursting (A) Membrane po- tential (top) and slow variables (mNaP, middle and hKas, bottom) through four cy- cles of bursting in the deterministic system. (B) Bifurcations in the fast subsys- tem overlayed on the model trajectory in the (mNaP, hKas)plane. The red dashed line indicates a subcritical Hopf bifurcation (SCH1), whereas the dotted red line indicates a homoclinic bifurcation (HC1). The black dashed line shows the lin- ear model that combines hKas and mNaP into a single slow variable, mslow. (C) Bifurcation analysis of the fast subsystem of the model using mslow as a bifurca- tion parameter. A stable equilibrium (solid black line) is shown to lose stability (dashed black line) via a subcritical Hopf bifurcation (SCH2). The stable periodic orbit (solid green line) disappears in a homoclinic bifurcation (HC2). A region of bistability exists (shaded region, zoomed in panel D). See text for a description of the remaning bifurcations. (D) A close up of the bifurcations occuring in the region of bistability shown in grey in panel C. The blue line indicates a trajectory of the full system through a single period of bursting, with arrows indicating the direction of time. Dashed and dotted red lines correspond to the bifurcations of the fast subsytem introduced in panel B.

Figure 2.11: Bifurcation diagram for mNaP for a single value of hKas (A) Full bifurcation diagram of the fast subsystem whilst hKas = 0.19. (B) Zoom in on the bistable region of A (shaded in grey) to demonstrate HC1 homoclinic and SCH1 subcritical Hopf bifurcations.

ends as the fast subsystem returns to the steady state branch.

Figure 2.10B suggests that the slow system can be reduced to a single slow variable mslow with the approximation mNaP = mslow and hKas = −0.7657mslow + 0.6477. This linear approximation of the two slow variables is shown in Fig- ure 2.10B. The full bifurcation diagram for the fast subsystem as mslow is varied is shown in Figure 2.10C. As before, the stable steady state is lost via subcrit- ical Hopf bifurcation (SCH2), and the stable periodic orbit is lost via homoclinic bifurcation (HC2). Figure 2.10C shows the remaining bifurcations. The unsta- ble periodic orbit generated by SCH2 is lost via a homoclinic (HC3). The un- stable steady state following SCH2 becomes stable via another subcritical Hopf (SCH3). The unstable periodic orbit generated by SCH3 collides with the stable periodic orbit generated in HC2 and both periodic orbits disappear via a saddle node of periodics (SNP1). As in the case of the two dimensional slow subsys- tem, there is bistability between the stable equilibrium and the stable periodic orbit (Figure 2.10D) resulting in traditional fast-slow hysteresis loop bursting. The trajectory of a single burst is shown inFigure 2.10D.