Mathematical Foundations of Neuroscience - Lecture 10. Bursting.
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1 Mathematical Foundations of Neuroscience - Lecture 10. Bursting. Filip Piękniewski Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland Winter 2009/2010 Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 1/49
2 Introduction Up to now, we ve seen that neurons may spike in response to input current We ve studied the geometrical mechanisms that lead from resting to spiking But this is not the end of the story. Some neurons, instead of sending one spike, send a whole bunch of spikes. These spike packets are called bursts. It is not yet obvious what is bursting for - it may be more reliable in information exchange. Burst may also encode a channel (frequency modulation) Today we will look into the mechanisms responsible for bursting Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 2/49
3 20 Neuron response Input current 0 20 V Time (ms) Figure: An example of a neuron bursting in response to a step current. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 3/49
4 Definition A burster is a dynamical system which for some values of parameters autonomously alternates between periodic orbit and a stable resting state. As we will see bursters can vary from very simple to very complex multidimensional monsters. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 4/49
5 How could such a behavior occur in 2d system? The only possibility is that the trajectory will follow a very awkward cycle resembling a hedgehog: However this seems strange, such a model would not resemble any neural models we ve seen! Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 5/49
6 Introduction ï0.2 ï0.4 ï0.6 ï0.8 ï1 ï2 ï1.5 ï1 ï dx = x x 3 /3 u + dt example of a planar burster. Figure: The system Filip Piękniewski, NCU Toruń, Poland cos(40u), du dt 1+e 5(1 x ) 3 = 0.01x is and Mathematical Foundations of Neuroscience - Lecture 10 6/49
7 Figure: Vector field of the hedgehog system represented as plains. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 7/49
8 In neuronal models however we do not see any hedgehogs! When the input current steps up, the phase portrait instantly changes and remains fixed. The trajectory can either converge back to equilibrium (possibly firing one spike), or remain on the limit cycle indefinitely. The neuron can be bistable, but still there has to be a current pulse that switches from one regime to the other. Note that bursting like activity may be induced by stimulating a neuron with sinusoidal input current. However in a real burster something else has to oscillate and modulate the spiking. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 8/49
9 0 10 Neuron response Input current V Time (ms) Figure: A sinusoidal current may elicit bursting like waveform ( I Na,p - I K bistable integrator) Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 9/49
10 Electrophysiology Introduction In neurons, there has to be some other mechanism responsible for spiking/quiescence alternation Spiking activates slow currents, which extinguish firing While the neuron remains quiet, the slow current deactivates, eventually destroying the stable equilibrium The neuron then fires another spike train, and so on... The exact neurophysiological mechanisms responsible for bursting can be of various types, voltage gated, calcium gated etc. The important thing to note, is that there is some slowly oscillating current modulating the fast spiking system. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 10/49
11 Fast-slow bursters Averaging Equivalent voltage One can add a slow potassium current to the I Na,p + I K model: C m dv dt = g L(V E L ) g Na m (V )(V E Na ) g K n(v E K )+ dn dt = (n (V ) n)/τ n (V ) dn slow = (n slow (V ) n slow )/τ slow (V ) dt g Kslow n slow (V E Kslow ) With C m = 1, E L = 80, τ(n) = g L = 8, g Na = 20, g K = 9, g Kslow = 5, E Na = 60, E K = 90, E Kslow = 90, τ slow (V ) = 20, m (V ) =, n 1+e 20 V (V ) =, n 15 1+e 25 V slow (V ) = 5 1+e 20 V 5 Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 11/49
12 Fast-slow bursters Averaging Equivalent voltage Neuron response Input current Slow current V Time (ms) Figure: I Na,p + I K + I K(M) bursting. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 12/49
13 Fast-slow bursters Averaging Equivalent voltage 10 Cycle manifold V Saddle node 70 bifurcation Saddle Saddle homoclinic bifurcation Resting state Slow current n Figure: Phase space of a burster (one of many possible). Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 13/49
14 Fast-slow bursters Introduction Fast-slow bursters Averaging Equivalent voltage A m + k fast-slow burster is a system which can be described as: d x dt = f ( x, u) d u dt = µg( x, u) where dim x = m, dim u = k and µ is small, such that the two equations operate on different time scales. The first equation is the fast subsystem while the second is the slow one. In neural models the fast subsystem is usually the 2d system responsible for spiking. The slow system can be 1d or 2d and is responsible for modulating spike trains and induces bursts. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 14/49
15 Dissecting a burster Introduction Fast-slow bursters Averaging Equivalent voltage Fast-slow bursters are convenient for analysis, since they can be dissected. One can freeze the slow dynamics and analyze the fast subsystem, as the slow variable drives it through various bifurcations On the other hand one can assume the fast system is instantaneous (similarly to instantaneous gating variables in the simplifications of Hodgkin-Huxley model) and study the behavior (phase portrait) of the slow system. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 15/49
16 Dissecting a burster Introduction Fast-slow bursters Averaging Equivalent voltage In order to dissect a burster d x dt = f ( x, u) d u dt = µg( x, u) we follow steps. If the neuron is resting, find the asymptotic resting value x rest for fixed u. We obtain the function x rest (u). We then study the reduced system: d u dt = µg( x rest(u), u) = g( u) If the neuron spikes (folows a periodic orbit) the asymptotic x (u) hasn t a fixed value. We can however replace the periodic function with an appropriate average. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 16/49
17 Dissecting a burster Introduction Fast-slow bursters Averaging Equivalent voltage We replace u with some other variable w = u + o(µ) and put g( w) = 1 n T ( w) n T ( w) 0 g ( x spike (t, w), w ) dt where T ( w) is the spiking period of the fast subsystem at w and x spike (t, w) is the value of the fast subsystem at t and n N is the number of cycles over which we compute the average (the more the better). Equilibria of the reduced slow system correspond to resting or spiking of the fast subsystem. Limit cycles may correspond to switching between spiking and resting (bursting behavior). The reduction fails for the points of transition from resting to spiking and vice versa (the period T ( w) may become infinite)! Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 17/49
18 Equivalent voltage Introduction Fast-slow bursters Averaging Equivalent voltage The reduced system is useful, but it looses the connection with the fast subsystem and therefore my be not very informative With the neural models the slow system usually depends only on voltage V, and not the whole vector of values V, n, m etc... One may solve g(v, u) = g( u) to find the equivalent voltage V equiv ( u). Note that V equiv ( u) = V rest ( u) when the neuron is resting. That way we can get back to a more familiar phase space of u with V attached. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 18/49
19 V Introduction Fast-slow bursters Averaging Equivalent voltage Neuron response Input current Slow current 30 V Spiking equivalent voltage Spiking amplitude Time (ms) n slow (V) saddle node n slow Stable resting state Figure: I Na,p + I K + I K(M) slow system phase portrait I = 3, 6, 10 Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 19/49
20 V Introduction Fast-slow bursters Averaging Equivalent voltage Neuron response Input current Slow current 30 V Spiking equivalent voltage Spiking amplitude Time (ms) n slow (V) saddle node Limit cycle (periodic bursting) Figure: I Na,p + I K + I K(M) slow system phase portrait I = 3, 6, 10 n slow Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 19/49
21 V Introduction Fast-slow bursters Averaging Equivalent voltage V Neuron response Input current Slow current 70 Spiking equivalent voltage Time (ms) Spiking amplitude n slow (V) saddle node Stable, persistent spiking n slow Figure: I Na,p + I K + I K(M) slow system phase portrait I = 3, 6, 10 Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 19/49
22 Fast-slow bursters Averaging Equivalent voltage Lets get back for a minute to the interpretation of bursting in which the slow subsystem drives the fast one through a series of bifurcations, which ultimately initiate and extinguish spiking. Note that the slow system has to exhibit periodic behavior. Therefore the minimal dimension for the slow subsystem has to be two. But we ve seen and bursters so far, how can they exist? The key is that the slow system is coupled with the fast one. If the fast system is bistable, it can drive the oscillations in the slow one via hysteresis loop. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 20/49
23 Fast-slow bursters Averaging Equivalent voltage Consider a simple system: dx = (x 1)x(x + 1) u dt du dt = 0.005x The fast system is bistable (stable states are positive and negative). When the fast system is at a positive state it slowly increases u, eventually reaching a point at which the equilibrium looses stability. At that point the system jumps to the negative equilibrium, and begins to decrease u. The process continues periodically, even though the slow system is 1d and cannot be periodic by itself. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 21/49
24 Fast-slow bursters Averaging Equivalent voltage Figure: Hysteresis loop in a pair of coupled 1d systems. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 22/49
25 Fast-slow bursters Averaging Equivalent voltage Figure: Hysteresis loop in a pair of coupled 1d systems. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 22/49
26 Fast-slow bursters Averaging Equivalent voltage Figure: Hysteresis loop in a pair of coupled 1d systems. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 22/49
27 Fast-slow bursters Averaging Equivalent voltage Figure: Hysteresis loop in a pair of coupled 1d systems. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 22/49
28 Fast-slow bursters Averaging Equivalent voltage Figure: Hysteresis loop in a pair of coupled 1d systems. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 22/49
29 Fast-slow bursters Averaging Equivalent voltage Figure: Hysteresis loop in a pair of coupled 1d systems. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 22/49
30 Fast-slow bursters Averaging Equivalent voltage Figure: Hysteresis loop in a pair of coupled 1d systems. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 22/49
31 Fast-slow bursters Averaging Equivalent voltage Figure: Hysteresis loop in a pair of coupled 1d systems. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 22/49
32 Fast-slow bursters Averaging Equivalent voltage Figure: Hysteresis loop in a pair of coupled 1d systems. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 22/49
33 Fast-slow bursters Averaging Equivalent voltage Figure: Hysteresis loop in a pair of coupled 1d systems. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 22/49
34 Fast-slow bursters Averaging Equivalent voltage Figure: Hysteresis loop in a pair of coupled 1d systems. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 22/49
35 Fast-slow bursters Averaging Equivalent voltage Figure: The slow subsystem (though 1d) exhibits periodic behavior due to hysteresis. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 23/49
36 Fast-slow bursters Averaging Equivalent voltage The same thing applies to bursters - when the fast system is bistable (resting/spiking) then 1d slow system is sufficient for bursting and can be driven by the hysteresis loop. If the fast system is monostable, the slow subsystem has to have at least two variables, giving four in total. Recall that Hodgkin-Huxley model is 4d, therefore it is a minimal model for bursting. The slow system even when it is 2d may not be able to oscillate by itself! The only coupling of the equations may be through the voltage. Recall that bistability in neural models appears near saddle-node homoclinic orbit bifurcation and Bautin bifurcation. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 24/49
37 Fast-slow bursters Averaging Equivalent voltage Stable equilibrium Stable cycle Bistability Figure: Bistability near saddle-node homoclinic orbit bifurcation Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 25/49
38 Fast-slow bursters Averaging Equivalent voltage saddle node on invariant circle circle/circle bursting circle/homoclinic bursting fold/circle bursting homoclinic saddle node fold/homoclinic bursting Figure: Possible ways that the slow system drives the fast one through bifurcations near saddle-node homoclinic orbit bifurcation. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 26/49
39 Fast-slow bursters Averaging Equivalent voltage Stable cycle Stable focus Unstable focus Bistability Figure: Bistability near Bautin bifurcation Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 27/49
40 Fast-slow bursters Averaging Equivalent voltage Fold cycle Subcritical AH a(b) subhopf/fold cycle bursting Bautin c(b) Bautin Supercritical AH subhopf/hopf bursting Hopf/Fold cycle bursting Hopf/Hopf bursting Figure: Possible ways that the slow system drives the fast one through bifurcations near Bautin bifurcation. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 28/49
41 Introduction Classification by E. Izhikevich Basic neurocomputational properties Any burst of spikes has two important stages - when it starts and when it ends. Each such an event corresponds to a bifurcation of the equilibrium and a limit cycle respectively. One can therefore classify bursters by the kind of bifurcation. Since we know there are four bifurcations of stable equilibria and for bifurcations of limit cycles, we end up with 16 possible bursters. Some of these bursters have been observed in real neurons, some are purely theoretical (perhaps they will be discovered in future). Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 29/49
42 Fold/Circle Bursting Fold/Homoclinic Bursting Fold/Hopf Bursting Fold/Fold Cycle Bursting Circle/Circle Bursting Circle/Homoclinic Bursting Circle/Hopf Bursting Circle/Fold Cycle Bursting Hopf/Circle Bursting SubHopf/Circle Bursting Hopf/Homoclinic Bursting SubHopf/Homoclinic Bursting Hopf/Hopf Bursting SubHopf/Hopf Bursting Hopf/Fold Cycle Bursting SubHopf/Fold Cycle Bursting Introduction Classification by E. Izhikevich Basic neurocomputational properties Spiking -> Resting Resting -> Spiking Saddle node on Invariant Circle Homoclinic Orbit Supercritical Andronov Hopf Fold Cycle Saddle node (Fold) Saddle node on Invariant Circle Supercritical Andronov Hopf Subcritical Andronov Hopf Figure: Diagram of major bursting types in neurons (see Izhikevich 2000). Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 30/49
43 Classification by E. Izhikevich Basic neurocomputational properties Fold/Circle Bursting Figure: Fold-Circle bursting diagram. The slow system requires 2 dimensions. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 31/49
44 Classification by E. Izhikevich Basic neurocomputational properties Circle/Circle Bursting Figure: Circle-Circle bursting diagram. The slow system requires 2 dimensions. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 32/49
45 Classification by E. Izhikevich Basic neurocomputational properties Hopf/Circle Bursting Figure: Supercritical Andronov-Hopf-Circle bursting diagram. The slow system requires 2 dimensions. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 33/49
46 Classification by E. Izhikevich Basic neurocomputational properties SubHopf/Circle Bursting Figure: Subcritical Andronov-Hopf-Circle bursting diagram. The slow system requires 2 dimensions. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 34/49
47 Classification by E. Izhikevich Basic neurocomputational properties Fold/Homoclinic Bursting Figure: Fold-Homoclinic bursting diagram. The slow system may be one dimensional due to hysteresis loop.. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 35/49
48 Classification by E. Izhikevich Basic neurocomputational properties Circle/Homoclinic Bursting Figure: Circle-Homoclinic bursting diagram. The slow system requires 2 dimensions. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 36/49
49 Classification by E. Izhikevich Basic neurocomputational properties Hopf/Homoclinic Bursting Figure: Supercritical Andronov-Hopf-Homoclinic bursting diagram. The slow system requires 2 dimensions. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 37/49
50 Classification by E. Izhikevich Basic neurocomputational properties SubHopf/Homoclinic Bursting Figure: Subcritical Andronov-Hopf-Homoclinic bursting diagram. The slow system requires 2 dimensions. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 38/49
51 Classification by E. Izhikevich Basic neurocomputational properties Fold/Hopf Bursting Figure: Fold-Supercritical Andronov-Hopf bursting diagram.the slow system requires 2 dimensions. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 39/49
52 Classification by E. Izhikevich Basic neurocomputational properties Circle/Hopf Bursting Figure: Circle-Supercritical Andronov-Hopf bursting diagram. The slow system requires 2 dimensions. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 40/49
53 Classification by E. Izhikevich Basic neurocomputational properties Hopf/Hopf Bursting Figure: Supercritical Andronov-Hopf-Supercritical Andronov-Hopf bursting diagram. The slow system requires 2 dimensions. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 41/49
54 Classification by E. Izhikevich Basic neurocomputational properties SubHopf/Hopf Bursting Figure: Subcritical Andronov-Hopf-Supercritical Andronov-Hopf bursting diagram. The slow system requires 2 dimensions. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 42/49
55 Classification by E. Izhikevich Basic neurocomputational properties Fold/Fold Cycle Bursting Figure: Fold-Fold Cycle bursting diagram. The slow system requires 2 dimensions. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 43/49
56 Classification by E. Izhikevich Basic neurocomputational properties Circle/Fold Cycle Bursting Figure: Circle-Fold Cycle bursting diagram. The slow system requires 2 dimensions. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 44/49
57 Classification by E. Izhikevich Basic neurocomputational properties Hopf/Fold Cycle Bursting Figure: Supercritical Andronov-Fold Cycle bursting diagram. The slow system requires 2 dimensions. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 45/49
58 Classification by E. Izhikevich Basic neurocomputational properties SubHopf/Fold Cycle Bursting Figure: Subcritical Andronov-Fold Cycle bursting diagram. The slow system may be one dimensional due to hysteresis loop. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 46/49
59 Introduction Classification by E. Izhikevich Basic neurocomputational properties in ik Sp Bistability at the end of the burst Bistability at the end of the burst g -> R R es es g tin tin g -> Sp Saddle node on Invariant Circle Supercritical Andronov Hopf Homoclinic Orbit Fold Cycle in ik g Bistability before the burst Fold/Circle Bursting Fold/Homoclinic Bursting Fold/Hopf Bursting Fold/Fold Cycle Bursting Saddle node (Fold) AaBb Circle/Homoclinic Bursting Circle/Circle Bursting Circle/Hopf Bursting Circle/Fold Cycle Bursting Saddle node on Invariant Circle Hopf/Circle Bursting Hopf/Homoclinic Bursting Hopf/Hopf Bursting Hopf/Fold Cycle Bursting SubHopf/Hopf Bursting SubHopf/Fold Cycle Bursting Supercritical Andronov Hopf SubHopf/Circle Bursting Bistability before the burst SubHopf/Homoclinic Bursting Subcritical Andronov Hopf AaBb Figure: Bistability at the beginning and end of the burst. Marked are models which are bistable during the whole burst (which can have 1d slow system). Are there any other bursters which can be bistable during the whole burst? Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 47/49
60 Introduction Classification by E. Izhikevich Basic neurocomputational properties in ik Sp Decreasing frequency at the end of the burst g Decreasing frequency at the end of the burst -> R R es es g tin tin g -> Sp Saddle node on Invariant Circle Homoclinic Orbit Supercritical Andronov Hopf Fold Cycle in ik g Fold/Circle Bursting Fold/Homoclinic Bursting Fold/Hopf Bursting Fold/Fold Cycle Bursting Increasing frequency at the beginning of the burst Saddle node (Fold) Circle/Circle Bursting Circle/Homoclinic Bursting Saddle node on Invariant Circle Circle/Hopf Bursting Circle/Fold Cycle Bursting AaBb Hopf/Circle Bursting Hopf/Homoclinic Bursting Hopf/Hopf Bursting Hopf/Fold Cycle Bursting SubHopf/Hopf Bursting SubHopf/Fold Cycle Bursting Supercritical Andronov Hopf SubHopf/Circle Bursting SubHopf/Homoclinic Bursting Subcritical Andronov Hopf Figure: Classification of bursters based on their spike frequency at the beginning and end of the burst. Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 48/49
61 itulation Introduction Bursting is a series of spikes alternating with a period of quiescence. In the general dynamical system sense, it is a system which autonomously alternates between period orbit and a resting state. Fast-slow bursters can be dissected into independent systems and studied theoretically Not all bursters are of the fast-slow type. These are very difficult to study. The slow system can be 1d (driven by the hysteresis loop - only when the fast system is bistable) or > 1d (slow wave type). Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 10 49/49
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