The Effects of Voltage Gated Gap. Networks

Transcription

1 The Effects of Voltage Gated Gap Junctions on Phase Locking in Neuronal Networks Tim Lewis Department of Mathematics, Graduate Group in Applied Mathematics (GGAM) University of California, Davis with Donald French Department of Mathematics, University of Cincinnati CNS*010 0Workshop: op Phase Locking in the Presence of Biological oogca Noise ose

2 gap unction/electrical synapse/electrical coupling Gap unction: Structure 1 Gap unction channel Connexin Connexon (hemi-channel) I coup, 1 gc V1 V ~ 0 different connexins

3 VOLTAGE GATING OF GAP JUNCTIONS C From Teubner et al (000) J. Membrane. Biol. 176: 49 6: Voltage gating of (homotypic) Cx36 gap unctions expressed in Xenopus Oocytes (A,B) and transfected HeLa cells (C). (A,C) initial and steady state normalized gap unction conductance (filled and open symbols respectively) as a function of transunctional potential V. (B) Representative unctional current for voltage clamp steps up to +10mV in 0mV increments. I coup, 1 gc V1 V voltage (and time ) dependent

4 goal: Explore the effects of gap unction voltage gating on phase locking in networks of coupled cells. Model: Pairs of oscillating spiking cells. Technique: Theory of weak coupled oscillators. Conditions: A. no voltage gating gating B. fast voltage gating C. slow voltage gating

5 MODEL 1 intrinsic dynamics: conductance based HH model e.g. fast spiking sp (FS) (S) interneuron eu model (Erisir et al 1999; Mancilla et al 007). Oscillatory for I 1 <I<I ; I controls the T period of the oscillations period T limit cycle oscillation C w dv dw ion I V, w w V w I V ( t) V 0 t, k 1,; k relative phase

6 MODEL 1 intrinsic dynamics: conductance based HH model e.g. fast spiking sp (FS) (S) interneuron eu model (Erisir et al 1999; Mancilla et al 007). Oscillatory for I 1 <I<I ; I controls the T period of the oscillations heterogeneity ( weak ) C w dv dw ion V, w I I 1 w V w I, k 1,; k

7 MODEL 1 intrinsic dynamics: conductance based HH model e.g. fast spiking sp (FS) (S) interneuron eu model (Erisir et al 1999; Mancilla et al 007). Oscillatory for I 1 <I<I ; I controls the T period of the oscillations noise ( weak ) C w dv dw I w ion V, w I t V w, k 1,; k

8 MODEL 1 intrinsic dynamics: conductance based HH model e.g. fast spiking sp (FS) (S) interneuron eu model (Erisir et al 1999; Mancilla et al 007). Oscillatory for I 1 <I<I ; I controls the T period of the oscillations heterogeneity ( weak ) coupling current ( weak ) C w dv dw ion V, w I g s V V I 1 w V w I c k, k 1,; k

9 MODEL Electrical coupling voltage gating dynamics: ds s ( V V ) s s k V -Vshift s ( V) 1-tanh s V scale scale *scaled to set s (0)=1 8 s V V shift shift 40mV, V 10mV, V scale scale 5mV 0mV V (mv)

10 THEORY OF WEAK COUPLING general: Malkin; Neu; Kuramoto; Ermentrout & Kopell; Hansel et al,, Schwemmer & Lewis electrical coupling: Lewis & Rinzel, Mancilla et al, Pfeuty et al,, Lewis & Skinner 1. Intrinsic i dynamics of the neurons dominate: ~ period T ~ intrinsic i i limit it cycle oscillation: V ( 0 ( t ) V0 t ( t ) slowly evolving relative phase

11 THEORY OF WEAK COUPLING. Response (phase shift) of a single cell to a small function like current stimulus: Infinitessimal phase resetting curve (iprc) Z(t): the normalized phase shift of a cell in response to a small function current stimulus of amplitude (total charge) I stim t that is delivered at phase t+. Z t I stim t

12 THEORY OF WEAK COUPLING 3. The effects of small continuously applied perturbations (coupling and heterogeneity) add sequentially according to the iprc t I t Z stim stim stim I t Z t t I t Z d k stim t I t Z,,

13 THEORY OF WEAK COUPLING 3. The effects of small continuously applied perturbations (coupling and heterogeneity) add sequentially according to the iprc d t I t, Z, Z stim k I t g s V t V t 1 c 0 k 0 phase equations coupling current ds s s V 0 ( t k ) V 0 ( t ) s heterogeneity k, 1; 1,; k

14 THEORY OF WEAK COUPLING 4. Averaging over an oscillation cycle: Average RHS of equations for the phases over period dtt to capture average rate of phase shift due to effect of the coupling and heterogeneity: d T 1 I Z( t ) gcsv0( tk) V0( t ) 1 T 0 T 1 I 1 Zt () gs c V0( t ( k) V0() t 1 Z() t T T 0 0 T effects of coupling current gc H k, s effects of heterogeneity 1 I I Q

15 A. NO VOLTAGE GATING i.e., constant s=s 0 =1 no T T d 1 I 1 Z () t g cs0 V0( t ( k) V0() t 1 Z() t T T 0 0 ( ) 1 g H c k I Q k, 1; 1,; k 1 = phase difference between cells d I I gch QgcH QgcHodd IQ gg IQ c

16 d gg c IQ Phase locking (according to the theory of weakly coupled oscillators) with I = 0: stable synchrony G() S AP S unstable anti phase

17 d gg c IQ The effects of heterogeneity (phase shifts and the loss of 1:1 phase locking): I Q/g c > G max G max 0 < I Q/g c < G max 0 I Q/g c =0 G() g c no phase locked states 1:1 phase locked states no phase locked states I ave I

18 Schematic of ArnoldTongues:

19 A. NO VOLTAGE GATING no 1 g c dv 1 C Iion ( V1, m1, h1, n1) Iapplied gc ( V V1) dv C Iion ( V, m, h, n1) Iapplied gc ( V1V) Intuition: electrical coupling promotes synchrony.

20 no PRCs and phase locking predictions for electrically coupled neurons S S/AP 50 Hz 5 Hz Z(t) V 0 (m PRC Z mv) phase t (msec) phase t (msec) G() phase difference (msec) phase difference (msec) *Mancilla, Lewis et al. (007) J. Neurosci.

21 no 50 Hz Synchrony Cell 1 Cell 50mV 100ms 5 Hz Synchrony 5 Hz Anti phase Cell 1 Cell 50mV 100ms *Mancilla, Lewis et al. (007) J. Neurosci.

22 no PRCs and phase locking predictions for electrically coupled neurons S S/AP 50 Hz 5 Hz Z(t) V 0 (m PRC Z mv) phase t (msec) phase t (msec) G() phase difference (msec) phase difference (msec) *Mancilla, Lewis et al. (007) J. Neurosci. Q: mechanism of antiphase? (see Lewis and Skinner review).

23 Q: What does voltage gating do to the synchronous and anti phase states?

24 FAST VOLTAGE GATING fast ds s V ( t ) V ( t ) s 0 s 0 k 0 0 s s V ( t ) V ( t ) 0 0 k 0 d T 1 0 I t g s V t V t V t V t 1 Z t 1 Z c 0 k 0 0 k T g c 0 H f 1 I Q k T T 0 d I I gc H f Qgc Hf Qgc Gf IQ

25 fast prediction from theory : voltage gating can promote anti phase activity. V 0 (mv) I 3.7 Acm / limit cycle oscillation s V V shift scale 40mV 5mV Z PRC V (mv) (msec)

26 fast prediction from theory : voltage gating can promote anti phase activity. no voltage gating voltage gating S AP S S AP S G () G f f () (msec) Without voltage gating, the theory predicts a stable synchronous activity (S) and a marginally stable anti phase state. (msec) With voltage gating, the synchronous state (S) and the anti phase state (AP) appear to be equally stable.

27 direct simulations fast g , I A / cm c I = 0.01A/cm voltage gating no voltage gating 50 mv 0 msec Without voltage gating, synchrony (S) was the only stable phase locked state observed for all initial conditions tried. With voltage gating, both synchrony (S) and anti phase (AP) activity were stable.

28 Phase model with (weak) additive white noise: d g c G I Q T t T T T Z ~ t ~ 0 1/ Fokker Planck equationfor distributionof of phase difference: c T,, t g G I Q, t t t (,t) ) = prob. that cells will have phase difference at time t.

29 effect of noise fast no voltage gating voltage gating S AP S S AP S () ) () ) (msec) (msec) Steady state cross correlogram () ) for two level of noise (solid low noise; dotted high noise ). is determined using the theory of weakly coupled oscillators (Pfeuty et al., 005): exp( ( )) g K where c 1 T 0 ( ), K ( ) G ( ) d, exp( K( )) d 0

30 C. SLOW VOLTAGE GATING slow For slow voltage gating, both the relative phases and s evolve on time scales much slower than the period T. Therefore, we can average their dynamics over the period T. T T d 1 I 1 Z() t g s V ( t( ) V () t 1 Z() t T T c 0 0 k ( ) 1 gs H c 0 k ds ( k ) s 0 I Q T 0 1 s s V0( t ( k) V0( t) s0 T 0 S d I I gs c 0 H Qgs c 0 H Q gs c 0 GIQ ds0 s S s0

31 predictions from theory : voltage gating always decreases the robustness of anti phase activity. slow I 3.0 A/ cm V 0 (m mv) limit cycle oscillation s V V shift scale 10mV 0mV PRC Z V (mv) (msec) S no voltage gating AP S G () (msec)

32 effect of heterogeneity : examining dynamics in the phase plane slow g c = 0 S S g c = 0. S s AP s AP (msec) (msec) d I gs c c 0 0GH IQ ds0 s S s 0 Q

33 effect of heterogeneity : examining dynamics in the phase plane slow g c = 0 S S g c = 0. S s AP s AP (msec) (msec) g c = g c = 4 S S s AP s (msec) (msec)

34 SUMMARY We have found a variety of effects that depend on the degree and type of voltage-gating as well as the intrinsic dynamics of the cells. Here, we include two examples of the effects of voltage-gated gated electrical coupling in pairs of spiking neurons: [1] When the voltage-gating process is fast, it can promote anti-phase activity. This is a result of a diminished synchronizing effect of the spikes. [] When the voltage-gating process is slow, it always promotes synchronous activity. Because it decreases the coupling strength only, slow rectification has no effect on the existence of phase-locked states in homogeneous, noiseless networks. However, it alters the robustness of these states. The synchronous state is relatively unaffected by the rectification, but the coupling strength can decrease substantially during asynchronous activity. Thus, the effective robustness of synchronous state increases in the presence of noise and heterogeneity.