Neuronal excitability from a dynamical systems perspective Mike Famulare CSE/NEUBEH 528 Lecture April 14, 2009

Mike Famulare, CSE/NEUBEH 5282 Outline Analysis of biophysical models –Hodgkin's classification of neurons by response to steady input currents Introduction to dynamical systems Phase portraits and some bifurcation theory –quadratic-integrate-and-fire model –Fitzhugh-Nagmuo model General simple neuron models

Mike Famulare, CSE/NEUBEH 5283 Biophysical Modeling Neurons can be modelled with a set of nonlinear differential equations (Hodgkin-Huxley)‏

Mike Famulare, CSE/NEUBEH 5284 Hodgkin-Huxley Model

Mike Famulare, CSE/NEUBEH 5285 What causes the spike in the HH model? In response to a step current at t=5 ms: –fast inward current followed by slower outward current –sodium channel (m) activates –more slowly, potassium channel (n) activates and sodium (h) deinactivates.

Mike Famulare, CSE/NEUBEH 5286 Hodgkin-Huxley f-I curve Rate coding: firing rate response (f) to input current (I), steady state There is a minimum firing rate (58 Hz)‏ Can you infer the f-I curve from the Hodgkin- Huxley equations?

Mike Famulare, CSE/NEUBEH 5287 Connor-Stevens Model Model of a neuron in anisodoris (AKA the sea lemon nudibranch)‏

Mike Famulare, CSE/NEUBEH 5288 What causes a Connor-Stevens spike?

Mike Famulare, CSE/NEUBEH 5289 Connor-Stevens f-I curve Does not have a minimum firing frequency.

Mike Famulare, CSE/NEUBEH Classifying Neurons by f-I type cortical pyramidal brainstem mesV Hodgkin's Classification of Neuronal Excitability Class 1: shows a continuous f-I curve (like Connor-Stevens)‏ Class 2: shows a discontinuous f-I curve (like Hodgkin-Huxley)‏ Class 3: shows no persistent firing (as can be found in auditory brainstem)‏

Mike Famulare, CSE/NEUBEH Model-by-model is not the way to go! We want to understand why neurons are excitable. We want to understand what makes different neurons behave in different ways. Going model-by-model is difficult and not at all general: –there are hundreds of channel types in nature –any cell expresses a few or ten or so of them What do we do with cells whose response is measurable but for which we don't have a model?

Mike Famulare, CSE/NEUBEH Dynamical systems and simple models The models we've talked about are dynamical systems. What's a dynamical system? We can analyze dynamical systems to understand: –equilibria (resting potentials)‏ –“unstable manifolds” (spike thresholds)‏ –bifurcations (f-I class, root of subthreshold behavior)‏

Mike Famulare, CSE/NEUBEH System: dynamical variables, control variable I. Fixed points : Linear response near a fixed point: Stability analysis: what are the eigenvalues of Bifurcation: change in the qualitative behavior of the system as the “control variable”, I, is changed. Overview of stability and bifurcation analysis

Mike Famulare, CSE/NEUBEH A note about the leaky-integrate-and-fire Leaky-integrate-and-fire (LIF) model: v o = resting potential, v r = reset voltage, v th = threshold voltage This model is not a spiking model in the sense that it doesn't have any dynamics for the spike itself. Piecewise “spike” is not dynamically similar to any real neuron Useful, but not for what we want to do today.

Mike Famulare, CSE/NEUBEH Quadratic-Integrate-and-Fire Model (QIF)‏ Simplest model with dynamical spikes:

Mike Famulare, CSE/NEUBEH QIF fixed points Fixed points: or Do the fixed point exist for all I? –for, the fixed points no longer exist (we'll come back to what this means soon!)

Mike Famulare, CSE/NEUBEH Phase plane analysis of the QIF: fixed points phase portrait for various external currents

Mike Famulare, CSE/NEUBEH Phase plane analysis of the QIF: stability Phase portrait of QIF: τ m =1, α=1, and I ext =0

Mike Famulare, CSE/NEUBEH Stability Analysis of QIF linear response and stability at each fixed point for v - –stable! v - is the resting potential for v + –unstable! v + is the threshold voltage

Mike Famulare, CSE/NEUBEH Saddle-node bifurcation in the QIF Loss of stability via a saddle-node bifurcation: – two fixed points “annihilate” each other also, we see that the QIF is an integrator!

Mike Famulare, CSE/NEUBEH Two types of saddle-node bifurcation “Saddle-node on invariant circle” (SNIC)‏ –reset below “ghost” of fixed point –arbitrarily low firing rate—Type I Saddle-node (SN)‏ –reset above “ghost” fp –slow first spike –finite minimum firing rate—Type II

Mike Famulare, CSE/NEUBEH Summary of saddle-node bifurcations Saddle-node bifurcations occur when two fixed points disappear in response to a changing input Systems showing an SN bifurcation will be act as integrators For neurons, depending on details of the nonlinear spike return mechanism, SN bifurcators can be Type I (continuous f-I curve) or Type II (discontinuous f-I curve)‏ The Connor-Stevens model shows a SNIC bifurcation

Mike Famulare, CSE/NEUBEH What about resonating neurons? The saddle-node bifurcation type is only one of the very simple (“codimension 1”) bifurcations Hodgkin-Huxley does not show a saddle node bifurcation –one of the many ways to see this is that the HH model cannot show an arbitrarily-long delayed first spike to step current With one dynamical variable, the saddle-node is the only possible continuous bifurcation, so we need two variables now!

Mike Famulare, CSE/NEUBEH Fitzhugh-Nagumo Model The Fitzhugh-Nagumo (FN) model is Hodgkin- Huxley like. Equations: Has two dynamical variables (is a two- dimensional dynamical system)‏ –a voltage variable, v –a recovery variable, w

Mike Famulare, CSE/NEUBEH Fitzhugh-Nagumo phase portrait For standard parameters, the FN has one fixed point that exists for all I. dx/dt = 0 lines are known as “nullclines”

Mike Famulare, CSE/NEUBEH Stability of the fixed point in the FN model finding the critical current –bifurcation happens when intersection of nullclines is at the local minimum: fixed point nearby critical point

Mike Famulare, CSE/NEUBEH Linear Response of FN near critical point Linear response of 2D model: eigenvalues are a complex-conjugate pair stable when

Mike Famulare, CSE/NEUBEH FN model shows a Hopf bifurcation Hopf bifurcation: stable, oscillatory fixed point becomes an unstable, oscillatory fixed point –there is a non-zero minimum firing rate controlled by the linear response frequency at the critical input current Type II excitability

Mike Famulare, CSE/NEUBEH Visualizing Hopf Dynamics phase portrait of a Hopf-bifurcating model (not the FN) for currents below the critical current looping around = subthreshold oscillation

Mike Famulare, CSE/NEUBEH Two types of Hopf bifurcation There are two types of Hopf bifurcation: –supercritical (like Fitzhugh-Nagumo and HH)‏ –subcritical –see “Dynamical Systems in Neuroscience” by Izhikevich or Scholarpedia for details For real single neurons, the difference has never been found to be experimentally important (Izhikevich 2007) Both are Type II

Mike Famulare, CSE/NEUBEH Bifurcation Theory Review Bifurcation: a change in the qualitative behavior in response to a changing control parameter With only one control parameter (e.g. current), there are only two types of equilibrium bifurcations (“codimension one”): There is a lot more to this bifurcation business! –what if you've got more inputs (drugs, hormones)? –how do you fit a simple model (“normal form”, ”canonical model”) to a more complex model or real data?

Mike Famulare, CSE/NEUBEH Simple Models can cover a lot of ground Saddle-node and Hopf bifurcations are very common and can describe the single-spike properties of the spike-generating mechanisms of most neurons One model to do a lot (Izhikevich 2003)‏

Mike Famulare, CSE/NEUBEH Izhikevich's simple model (adaptive QIF)‏

Mike Famulare, CSE/NEUBEH References Dayan and Abbott “Dynamical Systems in Neuroscience” by E.M. Izhikevich “Spiking Neuron Models” by Gerstner and Kistler “Nonlinear Dynamics and Chaos” by Strogatz Scholarpedia