Neuronal Dynamics: Computational Neuroscience of Single Neurons

Transcription

1 Nonlinear Integrate-and-Fire Model Neronal Dynamics: Comptational Neroscience of Single Nerons Nonlinear Integrate-and-fire (NLIF) - Definition - qadratic and expon. IF - Extracting NLIF model from data Week 1 and Week 4: Nonlinear Integrate-and-fire Model - exponential Integrate-and-fire - Extracting NLIF from detailed model - from two to one dimension Wlfram Gerstner EPFL, Lasanne, Switzerland - Qality of NLIF?

2 Neronal Dynamics Review: Nonlinear Integrate-and Fire LIF (Leaky integrate-and-fire) d ( ) RI( t) rest NLIF (nonlinear integrate-and-fire) d F( ) RI( t) If firing: reset

3 Neronal Dynamics 1.4. Leaky Integrate-and Fire revisited d I=0 LIF d I>0 d ( ) RI( t) rest If firing: r resting repetitive t t

4 Neronal Dynamics 1.4. Nonlinear Integrate-and Fire Nonlinear Integrate-and-Fire d I=0 d I>0 NLIF d F( ) RI( t) firing: if (t) = r then r r r

5 Nonlinear Integrate-and-fire Model j i Spike emission i r F reset I d F( ) RI( t) NONlinear if (t) = r then Fire+reset threshold

6 Nonlinear Integrate-and-fire Model d I=0 d I>0 r r d Qadratic I&F: F ( ) RI( t) NONlinear F ( ) ( ) c c c 0 if (t) = r then Fire+reset threshold

7 Nonlinear Integrate-and-fire Model d I=0 d I>0 r r d Qadratic I&F: F ( ) RI( t) F ( ) ( ) c c c 0 exponential I&F: if (t) = r then Fire+reset F ( ) ( ) c exp( 0 rest )

8 Nonlinear Integrate-and-Fire Model Neronal Dynamics: Comptational Neroscience of Single Nerons Nonlinear Integrate-and-fire (NLIF) - Definition - qadratic and expon. IF - Extracting NLIF model from data Week 1 and Week 4: Nonlinear Integrate-and-fire Model - exponential Integrate-and-fire - Extracting NLIF from detailed model - from two to one dimension Wlfram Gerstner EPFL, Lasanne, Switzerland

9 Neronal Dynamics Review: Nonlinear Integrate-and-fire See: week 1, lectre 1.5 r d f ( ) R I( t) If reset then reset to What is a good choice of f? r

10 Neronal Dynamics Review: Nonlinear Integrate-and-fire (1) d f ( ) R I( t) (2) If then reset to reset r What is a good choice of f? (i) Extract f from data (ii) Extract f from more complex models

11 Neronal Dynamics 1.5. Inject crrent record voltage

12 Neronal Dynamics Inject crrent record voltage voltage I(t) F ( ) ( ) exp( rest ) d 1 C I ( t) F( ) 1 [mv] Badel et al., J. Nerophysiology 2008

13 Neronal Dynamics Review: Nonlinear Integrate-and-fire (i) Extract f from data Badel et al. (2008) d f( ) f ( ) R I( t) f( ) d ( ) exp( ) rest Exp. Integrate-and-Fire, Forcad et al Pyramidal neron linear exponential Inhibitory interneron linear exponential Badel et al. (2008)

14 Neronal Dynamics Review: Nonlinear Integrate-and-fire (1) d f ( ) R I( t) (2) If then reset to reset r Best choice of f : linear + exponential d BUT: Limitations need to add -Adaptation on slower time scales ( ) exp( ) rest -Possibility for a diversity of firing patterns -Increased threshold after each spike -Noise

15 Week 4 part 5: Nonlinear Integrate-and-Fire Model Neronal Dynamics: Comptational Neroscience of Single Nerons Nonlinear Integrate-and-fire (NLIF) - Definition - qadratic and expon. IF - Extracting NLIF model from data Week 1 and Week 4: Nonlinear Integrate-and-fire Model - exponential Integrate-and-fire - Extracting NLIF from detailed model - from two to one dimension Wlfram Gerstner EPFL, Lasanne, Switzerland

16 Neronal Dynamics 4.5. Frther redction to 1 dimension After redction of HH 2-dimensional eqation stimls to two dimensions: d F(, w) RI( t) dw w G(, w) slow! Separation of time scales -w is nearly constant (most of the time)

17 Neronal Dynamics 4.5 sparse activity in vivo Spontaneos activity in vivo awake mose, cortex, freely whisking, -spikes are rare events Crochet et al., membrane potential flctates arond rest Aims of Modeling: - predict spike initation times - predict sbthreshold voltage

18 Neronal Dynamics 4.5. Frther redction to 1 dimension d F stimls (, w) I( t) w dw 0 w dw G(, w) Separation of time scales I(t)=0 w Flx nearly horizontal d 0 Stable fixed point

19 Neronal Dynamics 4.5. Frther redction to 1 dimension Hodgkin-Hxley redced to 2dim dw 0 w d dw F(, w) I( t) G(, w) Separation of time scales w d 0 dw w 0 w w rest d F(, w ) RI ( t) rest Stable fixed point

20 Neronal Dynamics Review: Nonlinear Integrate-and-fire (i) Extract f from more complex models d f ( ) R I( t) A. detect spike and reset resting state Separation of time scales: Arrows are nearly horizontal See week 3: d F(, w) R I( t) Spike initiation, from rest w w rest 2dim version of dw Hodgkin-Hxley w G(, w) B. Assme w=wrest

21 Neronal Dynamics Review: Nonlinear Integrate-and-fire (i) Extract f from more complex models d f ( ) R I( t) linear exponential See week 4: 2dim version of Hodgkin-Hxley d dw w F (, w ) R I ( t ) rest G(, w) w w rest Separation of time scales

22 Neronal Dynamics 4.5. Nonlinear Integrate-and-Fire Model d F(, w ) RI ( t) f ( ) RI ( t) rest Nonlinear I&F (see week 1!) Image: Neronal Dynamics, Gerstner et al., Cambridge Univ. Press (2014)

23 Neronal Dynamics 4.5. Nonlinear Integrate-and-Fire Model Exponential integrate-and-fire model (EIF) f ( ) ( ) exp( ) rest d F(, w ) RI ( t) f ( ) RI ( t) rest Nonlinear I&F (see week 1!) Image: Neronal Dynamics, Gerstner et al., Cambridge Univ. Press (2014)

24 Neronal Dynamics 4.5. Exponential Integrate-and-Fire Model Exponential integrate-and-fire model (EIF) f ( ) ( ) exp( ) rest linear Image: Neronal Dynamics, Gerstner et al., Cambridge Univ. Press (2014)

25 Neronal Dynamics 4.5. Exponential Integrate-and-Fire Model Direct derivation from Hodgkin-Hxley d C g m 3 h( E ) g n 4 ( E ) g ( E ) I( t) Na Na K K l l d C g [ m ( )] 3 h ( E ) g [ n ] 4 ( E ) g ( E ) I( t) Na 0 rest Na K rest K l l Forcad-Trocme et al, J. Nerosci f ( ) ( ) exp( ) rest d F(, h, n ) RI( t) f ( ) RI( t) rest rest gives expon. I&F

26 Neronal Dynamics 4.5. Nonlinear Integrate-and-Fire Model 2-dimensional eqation w d dw F(, w) RI( t) G(, w) Separation of time scales -w is constant (if not firing) Relevant dring spike and downswing of AP d f ( ) RI( t) threshold+reset for firing

27 Neronal Dynamics 4.5. Nonlinear Integrate-and-Fire Model 2-dimensional eqation w d dw F(, w) RI( t) G(, w) Separation of time scales -w is constant (if not firing) d f ( ) RI( t) Linear pls exponential

28 Nonlinear Integrate-and-Fire Model Neronal Dynamics: Comptational Neroscience of Single Nerons Nonlinear Integrate-and-fire (NLIF) - Definition - qadratic and expon. IF - Extracting NLIF model from data Week 1 and Week 4: Nonlinear Integrate-and-fire Model - exponential Integrate-and-fire - Extracting NLIF from detailed model - from two to one dimension Wlfram Gerstner EPFL, Lasanne, Switzerland - Qality of NLIF?

29 Neronal Dynamics 4.5 sparse activity in vivo Spontaneos activity in vivo awake mose, cortex, freely whisking, -spikes are rare events Crochet et al., membrane potential flctates arond rest Aims of Modeling: - predict spike initation times - predict sbthreshold voltage

30 Neronal Dynamics 4.5.How good are integrate-and-fire models? Badel et al., 2008 Aims: - predict spike initation times - predict sbthreshold voltage Add adaptation and refractoriness (week 7)

31 Neronal Dynamics Qiz 4.7. A. Exponential integrate-and-fire model. The model can be derived [ ] from a 2-dimensional model, assming that the axiliary variable w is constant. [ ] from the HH model, assming that the gating variables h and n are constant. [ ] from the HH model, assming that the gating variables m is constant. [ ] from the HH model, assming that the gating variables m is instantaneos. B. Reset. [ ] In a 2-dimensional model, the axiliary variable w is necessary to implement a reset of the voltage after a spike [ ] In a nonlinear integrate-and-fire model, the axiliary variable w is necessary to implement a reset of the voltage after a spike [ ] In a nonlinear integrate-and-fire model, a reset of the voltage after a spike is implemented algorithmically/explicitly

32 Neronal Dynamics Nonlinear Integrate-and-Fire Reading: W. Gerstner, W.M. Kistler, R. Nad and L. Paninski, Neronal Dynamics: from single nerons to networks and models of cognition. Chapter 4: Introdction. Cambridge Univ. Press, 2014 OR W. Gerstner and W.M. Kistler, Spiking Neron Models, Ch.3. Cambridge 2002 OR J. Rinzel and G.B. Ermentrot, (1989). Analysis of neronal excitability and oscillations. In Koch, C. Segev, I., editors, Methods in neronal modeling. MIT Press, Cambridge, MA. Selected references. -Ermentrot, G. B. (1996). Type I membranes, phase resetting crves, and synchrony. Neral Comptation, 8(5): Forcad-Trocme, N., Hansel, D., van Vreeswijk, C., and Brnel, N. (2003). How spike generation mechanisms determine the neronal response to flctating inpt. J. Neroscience, 23: Badel, L., Lefort, S., Berger, T., Petersen, C., Gerstner, W., and Richardson, M. (2008). Biological Cybernetics, 99(4-5): E.M. Izhikevich, Dynamical Systems in Neroscience, MIT Press (2007)