John Rinzel, x83308, Courant Rm 521, CNS Rm 1005 Eero Simoncelli, x83938, CNS Rm 1030

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1 Computational Modeling of Neuronal Systems (Advanced Topics in Mathematical Biology: G , G ) Tuesday, 9:30-11:20am, WWH Rm Prerequisites: familiarity with linear algebra, applied differential equations, statistics and probability. Grad credit: 3 points Course webpage: John Rinzel, rinzel@cns.nyu.edu, x83308, Courant Rm 521, CNS Rm 1005 Eero Simoncelli, eero@cns.nyu.edu, x83938, CNS Rm 1030 This course will focus on computational modeling of neuronal systems, from cellular to system level, from models of physiological mechanisms to more abstract models of information encoding and decoding. We will address the characterization of neuronal responses or identification of neuronal computations; how they evolve dynamically; how they are implemented in neural ware; and how they are manifested in human/animal behaviors. Modeling will involve deterministic and stochastic differential equations, information theory, and Bayesian estimation and decision theory. Lecturers from NYU working groups will present foundational material as well as current research. Examples will be from various neural contexts, including visual and auditory systems, decision-making, motor control, and learning and memory. Students will undertake a course project to simulate a neural system, or to compare a model to neural data. Abstract (April 4), written report and oral presentation (May 4). There may be occasional homework.

2 Computational Neuroscience What computations are done by a neural system? How are they done? WHAT? Feature detectors, eg visual system. Coincidence detection for sound localization. Memory storage. Code: firing rate, spike timing. Statistics of spike trains Information theory Decision theory Descriptive models HOW? Molecular & biophysical mechanisms at cell & synaptic levels firing properties, coupling. Subcircuits. System level.

3 Course Schedule, Spring 2010 Jan 19 Jan 26 Feb 2 Feb 9 Feb 16 Feb 23 Mar 2 Mar 9 Mar 16 Mar 23 Mar 30 Apr 6 Apr 13 Apr 20 Apr 27 Rinzel: Nonlinear neuronal dynamics I: cellular excitability and oscillations; case study, MSO coincidence detection Rinzel: Nonlinear neuronal dynamics II: networks; case studies. Simoncelli: Descriptive models of sensory encoding: LNP cascade, model fitting, integ&fire Shapley: V1 network models and gamma "oscillations" Rangan: Model for olfactory representation de la Rocha: Modeling the stochastic activity of recurrent balanced cortical circuits Tranchina: Population density methods and applications to gain control in neural networks. Reyes: Cascade modeling, auditory RF properties no class (Spring break) Pesaran: LFP, correlations between cortical areas Simoncelli: Sensory decoding: estimation/decision from neural populations Movshon: Visual motion representation, decision Rubin: Perceptual bistability Glimcher: Decision making, neuroeconomics Daw: Decision making and reinforcement learning

4 Nonlinear Dynamics of Neuronal Systems -- cellular level John Rinzel Computational Modeling of Neuronal Systems Spring 2010

5 References on Nonlinear Dynamics Rinzel & Ermentrout. Analysis of neural excitability and oscillations. In Koch & Segev (see below). Also as Meth3 on 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: Izhikevich, EM: Dynamical Systems in Neuroscience. The Geometry of Excitability and Bursting. MIT Press, Edelstein-Keshet, L. Mathematical Models in Biology. Random House, Strogatz, S. Nonlinear Dynamics and Chaos. Addison-Wesley, References on Modeling Neuronal System Dynamics Koch, C. Biophysics of Computation, Oxford Univ Press, Esp. Chap 7 Koch & Segev (eds): Methods in Neuronal Modeling, MIT Press, Wilson, HR. Spikes, Decisions and Actions, Oxford Univ Press, 1999.

6 Dynamics of Excitability and Oscillations Cellular level (spiking) Network level (firing rate) Hodgkin-Huxley model Membrane currents Wilson-Cowan model Activity functions Activity dynamics in the phase plane Response modes: Onset of repetitive activity (bifurcations)

7 Nonlinear Dynamical Response Properties Cellular: HH V(t) I app brief input brief input Type I, II Excitability Repetitive activity Bistability Bursting Network: Wilson-Cowan (Mean field) P exc P inhib 1 P exc 1 1 % firing P inhib brief input

8 Auditory brain stem neurons fire phasically, not to slow inputs. Blocking I KLT may convert to tonic. J Neurosci, 2002 Type III

9 Take Home Messages Excitability/Oscillations : fast autocatalysis + slower negative feedback Value of reduced models Time scales and dynamics Phase space geometry Different dynamic states Bifurcations ; concepts and methods are general. XPP software: (Bard Ermentrout s home page, then XPP )

10 Synaptic input many, O(10 3 to 10 4 ) Dendrites, 0.1 to 1 mm long Classical neuron Signals: V m ~ 100 mv membrane potential O(msec) ionic currents Axon, mm neurons, muscles Membrane with ion channels variable density over surface. Dendrites graded potentials, linear in classical view Axon characteristic impulses, propagation

11 Electrical Activity of Cells V = V(x,t), distribution within cell uniform or not?, propagation? Coupling to other cells Nonlinearities Time scales Current balance equation for membrane: V d 2 V C m t 4R x 2 i +I ion (V)= + I app + coupling capacitive channels cable properties other cells Coupling: g c,j (V j V) j electrical - gap junctions other cells I ion = I ion (V,W) g syn,j (V j (t)) (V syn -V) j = g k (V,W) (V V k ) k channel types generally nonlinear chemical synapses W/ t = G(V,W) gating dynamics

12 Nobel Prize, 1959

13 Current Balance: (no coupling, no cable properties, steady state ) 0 g K (V-V K ) + g Na (V-V Na ) + g L (V-V L ) gkvk + gnavna + glvl V g K + g Replace E by V Na + g L

14 HH Recipe: V-clamp I ion components Predict I-clamp behavior? I K (t) is monotonic; activation gate, n I Na (t) is transient; activation, m and inactivation, h e.g., g K (t) = I K (t) /(V-V K ) = G K n 4 (t) with V=V clamp gating kinetics: dn/dt = α(v) (1-n) β(v) n = (n (V) n)/τ n (V) n (V) increases with V. mass action for subunits or HH- particles OFF α(v) ON P P * β(v) I Na (t) = G Na m 3 (t) h(t) (V-V Na )

15 HH Equations C m dv/dt + G Na m 3 h (V-V Na ) + G K n 4 (V-V K ) +G L (V-V L ) = I app [+d/(4r) 2 V/ x 2 ] space-clamped dm/dt = φ [m (V)-m]/τ m (V) dh/dt = φ [h (V) - h]/τ h (V) dn/dt = φ [n (V) n]/τ n (V) φ, temperature correction factor = Q 10 **[(temp-temp ref )/10] HH: Q 10 =3 Reconstruct action potential Time course Velocity Threshold Refractory period Ion fluxes Repetitive firing? V

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17 HH model is a mean-field model that assumes an adequate density of channels. 1 μm 2 has about 100 Na + and K + channels.

18 Bullfrog sympathetic Ganglion B cell Cell is compact, electrically but not for diffusion Ca 2+ g c & g AHP depend on [Ca 2+ ] int MODEL: HH circuit + [Ca 2+ ] int + [K + ] ext Yamada, Koch, Adams 89

19 Cortical Pyramidal Neuron Complex dendritic branching Nonuniformly distributed channels Fall 2010: Math Aspects of Neurophys Pyramidal Neuron with axonal tree Koch, Douglas, Wehmeier 90 Schwark & Jones, 89

20 HH action potential biophysical time scales. I-V relations: I SS (V) I inst (V) steady state instantaneous HH: I SS (V) = G Na m 3 (V) h (V) (V-V Na ) + G K n 4 (V) (V-V K ) +G L (V-V L ) I inst (V) = G Na m 3 (V) h (V-V Na ) + G K n (V-V K ) +G L (V-V L ) fast slow, fixed at holding values e.g., rest

21 Dissection of HH Action Potential Fast/Slow Analysis - based on time scale differences h, n are slow relative to V,m Borisyuk A & Rinzel J, Chapt Idealize AP to 4 phases V h,n constant during upstroke and downstroke t C dv/dt = - I inst (V, m (V), h R, n R ) + I app Upstroke V,m slaved during plateau and recovery R and E stable T - unstable R T E

22 Repetitive Firing, HH model and others Response to current step nerve block Subthreshold

23 Repetitive firing in HH and squid axon -- bistability near onset Linear stability: eigenvalues of 4x4 matrix. For reduced model w/ m=m (V): stability if I inst / V + C m /τ n > 0. Interval of bistability Rinzel & Miller, 80 HH eqns Squid axon Guttman, Lewis & Rinzel, 80

24 Bistability in motoneurons

25 2-compartment model; plateau generator in dendrite Booth & Rinzel, 95 Morris-Lecar membrane model

26 Bistability in 2-compt model Response to up/down ramp; hysteresis Switching between states

27 Two-variable Model Phase Plane Analysis Reduction of HH, 2-variables: I Ca fast, non-inactivating Rinzel, 1985 I K -- delayed rectifier, like HH s I K Morris & Lecar, 81 barnacle musclel negative feedback: slow V K V L V Ca V rest V

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29 Get the Nullclines dv/dt = - I inst (V,w) + I app dw/dt = φ [ w (V) w] / τ w (V) w > w rest dv/dt = 0 I inst (V,w) = I app w= w rest w = w rest dw/dt = 0 w= w (V) rest state

30 ML model - excitable regime Case of small φ traj hugs V-nullcline - except for up/down jumps.

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32 Repetitive Activity in ML (& HH) Onset is via Hopf bifurcation Type II onset Hodgkin 48

33 Amplitude vs I app V max Bistability near onset - subcritical Hopf V min Frequency vs I app

34 I K - deactivated Anode Break Excitation or Post-Inhibtory Rebound (PIR)

35 Subthreshold nonlinearities: ipsp can enhance epsp, and lead to spiking Post inhibitory facilitation, PIF, transient form of PIR Model of coincidence-detecting cell in auditory brain stem. Has a subthreshold K + current I KLT.

36 Theory PIF Experiment (Gerbil MSO, slice) Competing factors: hyperpolar zn (farther from V th ) and hyperexcitable (reduced w) Dependence on τ inh Dodla, Svirskis, Rinzel - J Neurophys 2006

37 Boosting Spontaneous Rate with Fast Inhibition, via PIF Firing Response to Poisson-G ex train Enhanced by Inhibition (Poisson-G inh ) G ex only: Add G inh : # of spikes: 10, 15 Std HH model; 100 Hz inputs; τ ex = τ inh =1 ms w/ R Dodla. PhysRev E, 2006.

38 Adjust param s changes nullclines: case of 3 rest states 3 states I ss is N-shaped Stable or Unstable? 3 states not necessarily: stable unstable stable. Φ small enough, then both upper/middle unstable if on middle branch.

39 ML: φ large 2 stable steady states Neuron is bistable: plateau behavior. Saddle point, with stable and unstable manifolds V t e.g., HH with V K = 24 mv I app switching pulses

40 ML: φ small both upper states are unstable Neuron is excitable with strict threshold. saddle threshold separatrix V rest long Latency I ss must be N-shaped. I K-A can give long latency but not necessary.

41 Theta -model, Kopell and Ermentrout excitable Onset of Repetitive Firing 3 rest states SNIC- saddle-node on invariant circle I app w saddle-node V homoclinic orbit; infinite period limit cycle emerge w/ large amplitude zero frequency

42 ML: φ small Response/Bifurcation diagram freq ~ I I 1 low freq but no conductances very slow Firing frequency starts at 0. I K-A? (Connor et al 77) Type I onset Hodgkin 48

43 Firing rate model (Amari-Wilson-Cowan) for dynamics of excitatory-inhibitory populations. τ e dr e /dt = -r e + S e (a ee r e -a ei r i + I e ) τ i dr i /dt = -r i + S i (a ie r e a ii r i + I i ) r i (t), r e (t) -- average firing rate (across population and over spikes ) τ e, τ i -- recruitment time scale S e (input), S i (input) input/output relations, sigmoids a ee etc synaptic weights

44 Transition from Excitable to Oscillatory Type II, min freq 0 I ss monotonic subthreshold oscill ns excitable w/o distinct threshold excitable w/ finite latency Same for W-C network models. Type I, min freq = 0 I SS N-shaped 3 steady states w/o subthreshold oscillations excitable w/ all or none (saddle) threshold excitable w/ infinite latency Hodgkin 48 2 classes of repetiitive firing; Also - Class I less regular ISI near threshold

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46 Noise smooths the f-i relation frequency Type II Type I I app

47 FS cell near threshold RS cell, w/ noise FS cell, w/ noise

48 Take Home Message Excitability/Oscillations : fast autocatalysis + slower negative feedback Value of reduced models Time scales and dynamics Phase space geometry Different dynamic states Bifurcations XPP software: (Bard Ermentrout s home page)