Introduction to Mathematical Methods in Neurobiology: Dynamical Systems

1. Introductionto 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 Neuron as an Electric Circuit • 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.

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

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 Iapp=0 and fix w on a certain value, w0. • 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. Linearization of the Morris-Lecar Model • The Jacobian matrix is: • The eigenvalues are given by:

41. Linearization of the Morris-Lecar Model

42. Linearization of the Morris-Lecar Model

43. Linearization of the Morris-Lecar Model

44. 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 Iapp 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.