1. Introduction to Mathematical Methods in Neurobiology: Dynamical Systems Oren Shriki 2009 Neural Excitability

3. References about neurons as electrical circuits: Koch, C. Biophysics of Computation, Oxford Univ. Press, 1998. Tuckwell, HC. Introduction to Theoretical Neurobiology, I&II, Cambridge UP, 1988.

4. References about neural excitability: Rinzel & Ermentrout. Analysis of neural excitability and oscillations. In, Koch & Segev (eds): Methods in Neuronal Modeling, MIT Press, 1998. Also as “Meth3” on www.pitt.edu/~phase/ Borisyuk A & Rinzel J. Understanding neuronal dynamics by geometrical dissection of minimal models. In, Chow et al, eds: Models and Methods in Neurophysics (Les Houches Summer School 2003), Elsevier, 2005: 19-72. Izhikevich, EM. Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. MIT Press, (soon).

5. Neural Excitability One of the main properties of neurons is that they can generate action potentials. In the language of non-linear dynamics we call such systems “excitable systems”. The first mathematical model of a neuron that can generate action potentials was proposed by Hodgkin and Huxley in 1952. Today there are many models for neurons of diverse types, but they all have common properties.

6. A Typical Neuron

7. Intracellular Recording

8. Electrical Activity of a Pyramidal Neuron

9. The starting point in models of neuronal excitability is describing the neuron as an electrical circuit. The model consists of a dynamical equation for the voltage, which is coupled to equations describing the ionic conductances. The non-linear behavior of the conductances is responsible for the generation of action potentials and determines the properties of the neuron. The Neuron as an Electric Circuit

10. Hodgkin & Huxley Model (1952) The model considers an isopotential membrane patch (or a “point neuron”). It contains two voltage-gated ionic currents, K and Na, and a constant- conductance leak current.

11. The Equivalent Circuit External solution Internal solution

12. K and Na Currents During an Action Potential

13. Positive and Negative Feedback Cycles During an Action Potential

14. The Hodgkin-Huxley Equations

15. The Hodgkin & Huxley Framework Each gating variable obeys the following dynamics: - Represents the effect of temperature - Time constant

16. The Hodgkin & Huxley Framework The current through each channel has the form: - Maximal conductance (when all channels are open) - Fraction of open channels (can depend on several W variables).

17. The Temperature Parameter Φ Allows for taking into account different temperatures. Increasing the temperature accelerates the kinetics of the underlying processes. However, increasing the temperature does not necessarily increase the excitability. Both increasing and decreasing the temperature can cause the neuron to stop firing. A phenomenological model for Φ is:

18. Simplified Versions of the HH Model Models that generate action potentials can be constructed with fewer dynamic variables. These models are more amenable for analysis and are useful for learning the basic principles of neuronal excitability. We will focus on the model developed by Morris and Lecar.

19. The Morris-Lecar Model (1981) Developed for studying the barnacle muscle. Model equations:

20. Morris-Lecar Model The model contains K and Ca currents. The variable w represents the fraction of open K channels. The Ca conductance is assumed to behave in an instantaneous manner.

21. Morris-Lecar Model A set of parameters for example:

22. Morris-Lecar Model Voltage dependence of the various parameters (at long times):

23. Nullclines We first analyze the nullclines of the model (the curves on which the time derivative is zero). The voltage equation gives: To see what this curve looks like, let us assume that I app =0 and fix w on a certain value, w 0. This is the instantaneous I-V relation of the system:

24. Nullclines This curve is N-shaped and increasing w approximately moves the curve upward: We are interested in the zeros of this curve for each value of w.

25. Nullclines The nullcline of w is simply the activation curve:

26. Nullclines The nullclines in the phase plane:

27. Nullclines The steady state solution of the system is the point where the nullclines intersect. In the presence of an applied current, the steady-state satisfies:

28. Steady-State Current

29. General Properties of Nullclines The nullclines segregate the phase plane into regions with different directions for a trajectory’s vector flow. A solution trajectory that crosses a nullcline does so either vertically or horizontally. The nullclines’ intersections are the steady states or fixed points of the system.

30. The Excitable Regime We say that the system is in the excitable regime when the following conditions are met: –It has just one stable steady-state in the absence of an external current. –An action potential is generated after a large enough brief current pulse.

31. Effect of Brief Pulses on the ML Neuron A brief current pulse is equivalent to setting the initial voltage at some value above the rest state. Below are responses for the following initial voltages: -20, -12, -10, 0 mV

32. Order of Events If the pulse is large enough, autocatalysis starts, the solution’s trajectory heads rightward toward the V-nullcline. After the trajectory crosses the V -nullcline (vertically, of course) it follows upward along the nullcline’s right branch. After passing above the knee it heads rapidly leftward (downstroke).

33. Order of Events The number of K+ channels open reaches a maximum during the downstroke, as the w-nullcline is crossed. Then the trajectory crosses the V -nullcline’s left branch (minimum of V) and heads downward, returning to rest (recovery).

34. Dependence of Action Potential Amplitude on the Initial Conditions The amplitude of the action potential depends on the initial conditions. This contradicts the traditional view that the action potential is an all-or-none event with a fixed amplitude. If we plot the peak V vs. the size of the pulse or initial condition V0, we get a continuous curve. The curve becomes steep only at low temperature ( Φ ).

35. Dependence of Action Potential Amplitude on the Initial Conditions For small Φ, w is very slow and the flow in the phase plane is close to horizontal, since V is relatively much faster than w: In this case, it takes fine tuning of the initial conditions to evoke the graded responses.

36. Dependence of Action Potential Amplitude on the Initial Conditions The analysis above suggests a surprising prediction: If the experimentally observed action potentials look like all-or-none events, they may become graded if recordings are made at higher temperatures. This experiment was suggested by FitzHugh and carried out by Cole et al. in 1970. It was found that if recordings in squid giant axon are made at 38◦C instead of, say, 15◦C then action potentials do not behave in an all-or none manner.

37. Repetitive Firing at the ML Model In response to prolonged stimulation the cell fires spikes in a repetitive manner. From a dynamical point of view, the rest state is no longer stable and the system starts to oscillate. Typically, when the applied current is increased, the firing rate increases as well. We will examine to cases: –Generation of oscillations with a non-zero frequency –Generation of oscillations with a zero frequency

38. Linearization of the Morris-Lecar Model At what current will the steady state loose stability? To answer this, we will use linear stability theory. The model equations are:

39. Linearization of the Morris-Lecar Model Denote the steady state by: Consider the effects of a small perturbation: Using linearization we obtain:

40. The Jacobian matrix is: The eigenvalues are given by: Linearization of the Morris-Lecar Model

45. Onset of Oscillations For the parameter values we are using the eigenvalues have a negative real part. There are two options for loosing stability: Case 1: One of the eigenvalues becomes zero. In this case detJ=0 at the bifurcation.(Saddle node bifurcation). Case 2: The real part of both eigenvalues becomes zero at the bifurcation. After the bifurcation there are two complex eigenvalues with positive real part. In this case, TrJ=0 at the bifurcation.(Hopf bifurcation).

46. Onset of Oscillations with Finite Frequency The determinant is given by: When substituting the steady-state values we obtain: For these parameters values the steady-state I-V curve is monotonic and the loss of stability can happen only through Hopf bifurcation.

47. Hopf Bifurcation After the bifurcation the trace is positive: This condition can be interpreted as the autocatalysis rate being faster than the negative feedback’s rate. When this condition is met we obtain a limit cycle with a finite frequency. The frequency of the repetitive firing is proportional to the imaginary part of the eigenvalues.

48. Hopf Bifurcation We can estimate the current at which repetitive firing will start by plotting TrJ as a function of the applied current and estimating when it is zero. We will do it numerically. For each value of the applied current we draw the nullclines and estimate the steady-state values of V and w (the intersection of the nullclines):

49. Nullclines for Different Values of I app I

50. Hopf Bifurcation We then estimate TrJ and plot it as a function of the applied current: The critical current is around I=100.

51. Hopf Bifurcation We can also estimate the frequency: Around I=100 the imaginary part is close to 0.084. The frequency should be around: 0.084/(2*pi)*1000 =13.4 Hz

52. f-I Curve A plot of the f-I curve, obtained using numerical simulations of the dynamics, confirms our analysis:

53. Bifurcation Diagram The full bifurcation diagram is: The system experiences a subcritical Hopf bifurcation.

54. The Effect of the Applied Current on the Nullclines and the State of the Neuron At low currents the neuron is excitable. For intermediate currents it presents oscillations. At high currents the neuron experiences nerve block – there is no firing but the membrane is depolarized. The K and Ca currents balance each other.

55. Three Steady-States We next examine the model with the same parameters as before, except: In this case, the steady-state I-V curve becomes N- shaped. Therefore, for some values of the applied current there are three steady-states. The number and location of the steady-states do not depend on the temperature ( Φ ). However, the stability and the onset of repetitive firing do depend on Φ.

56. Steady-State Current

57. Large Φ - Bistability of Steady-States At large temperature ( Φ ) the rate of the w is not slow enough and there are no oscillations. The variable w can be considered instantaneous, w=w  (V). The model is then reduced to one dynamic variable, V, which simplifies the analysis. We will examine the model with Φ =1.

58. Large Φ - Bistability of Steady-States The middle steady-state is unstable and the upper and lower ones are stable. The neuron presents bistability between two steady- states. The voltage can be switched from one constant level to the other by brief pulses (“plateau behavior”).

59. Small Φ - Onset of Repetitive Firing at Zero Frequency We now consider the small temperature case, and examine the model with Φ = 0.06666667. The phase-plane picture: The two unstable manifold branches form (topologically) a circle that has two fixed- points on it. As the applied current is increased the two points coalesce and disappear. At this point a, limit cycle with infinite period is formed.

60. Saddle-Node Bifurcation The system experiences a saddle-node bifurcation. The bifurcation can be illustrated schematically in the following manner: After the bifurcation, the trajectory slows down near the “ghost” of the stable steady-state. This way, an arbitrarily slow frequency can be obtained.

61. Saddle-Node Bifurcation Near the bifurcation the frequency is proportional to the square root of the deviation from the critical current: The f-I curve:

62. Saddle-Node Bifurcation Voltage vs. time close to the bifurcation (I app =40):

63. Bifurcation Diagram

64. Intermediate Φ – Bistability of Rest State and a Depolarized Oscillation At intermediate values of Φ the upper steady state is surrounded by a stable periodic orbit. The phase-plane picture for Φ = 0.25 : In this case, there is bistability between the low- voltage stationary point and depolarized repetitive firing.

65. Intermediate Φ – Bistability of Rest State and a Depolarized Oscillation Brief current pulses can be used to switch between the states:

66. Repetitive Firing – Types of Excitability We can now summarize the characteristic properties of two generic types of transition from the excitable to oscillatory mode in neuronal models: Type II Type I MonotonicN-shapedI ss YesNoSubthreshold oscillations No distinct threshold. Potentials can be graded All-or-none excitability with strict threshold Threshold FiniteZeroFrequency at bifurcation

67. Repetitive Firing – Types of Excitability The two cases have different f-I relations. In type II the frequency jumps to finite values at the bifurcation. In type I the f-I curve is continuous and near the bifurcation there is a universal square-root dependence. However, if there is noise in the system, the f-I curves for both types will become qualitatively similar. The discontinuity of type II will disappear with noise since the probability of firing becomes zero where it was zero before. Type I’s f-I curve will also inherit a smooth foot rather than the infinite slope at zero frequency.

68. Types of Excitability in the ML Model The first set of parameters we studied exhibited type II excitability. The second set of parameters we studied exhibited type I excitability.

69. Neuronal Excitability – Characteristics of the Non-Linear Dynamics Excitability – the system responds to a large enough brief pulse with an action potential. Repetitive firing – the system exhibits oscillations in response to prolonged stimulation.

70. Neuronal Excitability – Main Properties of Models Autocatalysis – e.g. –The Na current in the HH model –The Ca current in the ML model Slow (enough) negative feedback– e.g. –K current –Inactivation of Na current in the HH model

71. Back to the Hodgkin-Huxley Model The First Intracellular Recording from the Squid Axon

72. The Equivalent Circuit External solution Internal solution

73. Hodgkin & Huxley Model

75. Ionic Conductances During an Action Potential

76. Repetitive Firing through Hopf Bifurcation in Hodgkin–Huxley Model A: Voltage time courses in response to a step of constant depolarizing current. from bottom to top: Iapp= 5, 15, 50, 100, 200 in μamp/cm2). Scale bar is 10 msec. B: f-I curves for temperatures of 6.3,18.5, 26◦C, as marked. Dotted curves show frequency of the unstable periodic orbits.

77. Repetitive Firing through Hopf Bifurcation in Hodgkin–Huxley Model With the standard parameters, the HH model exhibits type II excitability. The bifurcation is a Hopf bifurcation. In different parameter regimes the model can also exhibit type I excitability.

78. Fast-Slow Dissection of the Action Potential n and h are slow compared to m and V. Based on this observation, the system can be dissected into two time-scales. This simplifies the analysis. For details see: Borisyuk A & Rinzel J. Understanding neuronal dynamics by geometrical dissection of minimal models. In, Chow et al, eds: Models and Methods in Neurophysics (Les Houches Summer School 2003), Elsevier, 2005: 19-72.

79. Correlation between n and h During the action potential the variables n and h vary together. Using this correlation one can construct a reduced model. The first to observe this was Fitzhouh.

80. Conductance-Based Models of Cortical Neurons

81. Cortical neurons behave differently than the squid axon that Hodgkin and Huxley investigated. Over the years, people developed several variations of the HH model that are more appropriate for describing cortical neurons. We will now see an example of a simple model which will later be useful in network simulations. The model was developed by playing with the parameters such that it has an f-I curve similar to that of cortical neurons.

82. Frequency-Current Responses of Cortical Neurons Excitatory Neuron: Ahmed et. al., Cerebral Cortex 8, 462-476, 1998 Inhibitory Neurons: Azouz et. al., Cerebral Cortex 7, 534-545, 1997

83. Frequency-Current Responses of Cortical Neurons The experimental findings show what f-I curves of cortical neurons are: Continuous – starting from zero frequency. Semi-Linear – above the threshold current the curve is linear on a wide range. How can we reconstruct this behavior in a model?

84. An HH Neuron with a Linear f-I Curve Shriki et al., Neural Computation 15, 1809–1841 (2003)

85. Linearization of the f-I Curve We start with an HH neuron that has a continuous f-I curve (type I, saddle-node bifurcation). The linearization is made possible by the addition of a certain K-current called A-current. The curve becomes linear only when the time constant of the A-current is slow enough (~20 msec). There are other mechanisms for linearizing f-I curves.

86. Model Equations: Shriki et al., Neural Computation 15, 1809–1841 (2003)

87. Bursting at the Cellular Level Bursting is an oscillation mode with relatively slow rhythmic alternations between an active phase of rapid spiking and a phase of quiescence. Since its early discovery bursting has been found as a primary mode of behavior in many nerve and endocrine cells. Bursting can be a single-cell property or a network property. We will now focus on a simple example of a bursting mechanism at the cellular level by extending the Morris- Lecar model.

88. ML Model with Ca-dependent K-current The model is based on the ML model with the set of parameters that gives bistability of a rest state with a depolarized limit cycle (intermediate Φ ). The Ca-dependent K-current is described by:

89. ML Model with Ca-dependent K-current During a series of action potentials Ca concentration slowly builds up and increases the Ca-dependent K- current. When the Ca-dependent K-current is large enough, it brings the voltage down, terminating the active phase. This specific mechanism is termed “square wave” bursting.

90. Fast-Slow Dissection of Bursting Dynamics A general framework for analysis of bursting dynamics is to separate all dynamic variables into two subgroup: “fast” (X) and “slow (Y). In the context of bursting analysis a variable is considered fast if it changes significantly during an action potential, and slow if it only changes significantly on the time scale of burst duration. The general equations have the form:

91. Fast-Slow Dissection of Bursting Dynamics The analysis of the system can be conducted in two steps: Step 1. First we think of the slow variables as parameters and describe the spike generating fast subsystem for X as a function of Y. This description involves finding steady states, oscillatory orbits and their periods, and transitions between all these solutions (bifurcations) as a function of Y. Step 2. To describe the full system we overlay the slow dynamics on the fast system behavior. Y evolves slowly in time according to its equations, while X is tracking its stable states. Therefore, we must understand the direction of change of Y at each part of the bifurcation diagram for X.