Bifurcation examples in neuronal models

Bifurcation examples in neuronal models Romain Veltz / Olivier Faugeras October 15th 214

Outline Most figures from textbook of Izhikevich 1 Codim 1 bifurcations of equilibria 2 Codim 1 bifurcations of limit cycles 3 Additional bifurcations examples (3d) 4 Codim 2 bifurcation: Bogdanov-Takens 5 Excitability 6 Integrators / Resonators 7 Summary

Reminder on the I Na,p + I K model This is a simplified 2-dimensional simplification of the Hodgkin-Huxley model where we assume where a leak current a persistent Na-current with instantaneous kinetics a persistent K-current with slower kinetics C dv dt dn dt = I = n (V ) n τ(v ) I K {}}{ ḡ K n(v E K ) (m, n) = I Na {}}{ ḡ Na m (V )(V E Na ) g L (V E L ) 1 1 + exp[(v 1/2 V )/k]

1 ms 4 examples of codim 1 bifurcations Figures from Izhikevich saddle-node on invariant circle bifurcation 2 mv 1 ms Bifurcations -6 mv 6 pa saddle-node bifurcation supercritical Andronov-Hopf bifurcation 5 ms 1 ms saddle-node on invariant circle bifurcation subcritical Andronov-Hopf bifurcation 2 mv 1 ms 2 mv 1 ms -6 mv 6 pa -6 mv 2 na 5 ms supercritical Andronov-Hopf bifurcation Figure 6.1: Transitions from resting to tonic (periodic) spiking occur via bifurcatio equilibrium (marked by arrows). Saddle-node on invariant circle bifurcation: in recording of pyramidal neuron of rat s primary visual cortex. Subcritical Andr Hopf bifurcation: in vitro recording of brainstem mesencephalic V neuron. The two traces are simulations of the INa,p+IK-model.

I-V curves in the I Na,p + I K model 162 Bifurcations saddle-node bifurcation Andnronov-Hopf bifurcation rest state I(V) rest state I(V) no equilibria new equilibrium I= I> I= I> -1-5 membrane voltage, V (mv) (a) -1-5 membrane voltage, V (mv) (b) Figure 6.3: Steady-state I-V curves of the I Na,p +I K -model with high-threshold (left) and low-threshold (right) K + current (parameters as in Fig.4.1).

Neural properties of bifurcations Quasi-static change in I, bifurcations? Bifurcations 161 Fast Bifurcation of an equilibrium subthreshold Amplitude Frequency oscillations of spikes of spikes saddle-node no nonzero nonzero saddle-node on invariant circle no nonzero A I I b supercritical Andronov-Hopf yes A I I b nonzero subcritical Andronov-Hopf yes nonzero nonzero Figure 6.2: Summary of codimension-1 bifurcations of an equilibrium. I denotes the amplitude of the injected current, I b is the bifurcation value, and A is a parameter that depends on the biophysical details.

Saddle-node bifurcation 1/2 Figures from Izhikevitch, I Na,p + I K model with high threshold, τ(v ) =.152 166 Bifurc fast K+ current Bifurcations 165 (a) saddle-node bifurcation node saddle saddle-node (b) saddle-node on invariant circle (SNIC) bifurcation limit invariant circle cycle K + activation variable, n.6.5.4.3.2.1 n-nullcline V-nullcline saddle-node node saddle saddle-node.6 Figure 6.6: Two types of saddle-node bifurcation. originating and terminating at the same point. When the point disappears, the circle becomes a limit cycle. Both types of the bifurcation can occur in the INa,p+IK-model, as we show in Fig.6.7. The difference between the top and the bottom of the figure is the time, n.5.4 slow K+ current n-nullcline

Saddle-node bifurcation 2/2 the bifurcation occurs at I p = 4.51, (V p, n p ) = ( 61,.7) We can compute the center manifold we check that We have the normal form a = 1 2 f VV (V p, I p ) c = f O (V p, I p ) V = c(i I p ) + a(v V p ) 2 a =.1887, c = 1

Saddle-node bifurcation on limit cycle 1/2 Figures from Izhikevitch, I Na,p + I K model with high threshold, τ(v ) = 1 K + activation variable, n.4.3.2 V-nullcline.1 Also called Saddle-node homoclinic bifurcation ifurcations 165 (a) saddle-node bifurcation limit cycle.6.5 node saddle saddle-node slow K+ current saddle-node n-nullcline (b) saddle-node on invariant circle (SNIC) bifurcation invariant circle node saddle saddle-node Figure 6.6: Two types of saddle-node bifurcation. K + activation variable, n.4.3.2.1 V-nullcline invariant circle riginating and terminating at the same point. When the point disappears, the circle ecomes a limit cycle. Both types of the bifurcation can occur in the INa,p+IK-model, as we show in ig.6.7. The difference between the top and the bottom of the figure is the time onstant τ(v ) of the K + current. Since the K + current has a high threshold, the ime constant does not affect dynamics at rest, but it makes a huge difference when n action potential is generated. If the current is fast (top), it activates during the pstroke, thereby decreasing the amplitude of the action potential, and deactivates uring the downstroke, thereby resulting in overshoot and another action potential. n contrast, the slower K + current (bottom) does not have time to deactivate during he downstroke, thereby resulting in undershoot (short afterhyperpolarization), with going below the resting state. saddle-node on invariant circle -8-7 -6-5 -4-3 -2-1 1 2 membrane voltage, V (mv) Figure 6.7: Saddle-node bifurcation in the I Na,p+I K-model with a high-thresh current can be off the limit cycle (top) or on the invariant circle (bottom). Par are as in Fig.4.1a with τ(v )=.152 (top) or τ(v )=1(bottom).

-8 Saddle-node bifurcation on limit cycle 2/2 Saddle-node homoclinic bifurcation Bifurcations 167 Time from A to B π T 2 ac(i Ip) Action potential duration T 1 4.7ms Firing frequency ω = 1 T 1 +T 2 K + gating variable, n.5.4.3.2.1 V-nullcline n-nullcline A 1 8 6 4 2 frequency, Hz 4.5 5 5.5 injected dc-current, I B 1/T2 theoretical numerical -8-7 -6-5 -4-3 -2-1 1 2 membrane voltage, V (mv) -3 cut spike cut spike membrane voltage, V (mv) -4-5 -6-7 B A tan t

-1 supercritical -2 Andronov-Hopf Supercritical Andronov-Hopf bifurcation -3 bifurcation max/min of oscillations of membrane potential, mv -4-5 -6 stable stable cycles unstable equilibria -7-8 2 4 6 8 injected DC current, I I=2 I=25 urcations 171 I=15 I=1 n V stable limit cycles I=4 I= stable unstable I I=3 I=-1 max/min of oscillations of membrane potential, mv -1-2 -3-4 -5-6 stable cycles supercritical Andronov-Hopf bifurcation stable unstable equilibria -7-8 2 4 6 8 injected DC current, I I=15 I=2 I=25.7.6.5.4.3.2.1 V-nullcline n-nullcline imaginary part, ω supercritical Andronov-Hopf bifurcation at I=14.66 C eigenvalues c + iω I=1-8 -6-4 -2 2 membrane voltage, V (mv) stable unstable real part, c I=-1.7 I= Figure 6.11: Supercritical Andronov-Hopf bifurcation in the INa,p+IK-model with threshold K + current: As the bifurcation parameter I increases, the equilibrium stability and gives birth to a stable limit cycle with growing amplitude. Param are as in Fig.4.1b..6.5.4 V-nullcline supercritical Andronov-Hopf bifurcation at I=14.66 C eigenvalues c + iω.3 part, ω

Supercritical Andronov-Hopf bifurcation I Na,p + I K model with low threshold 172 Bifurcations 3 2 1 eigenvalues = c(i) + ω(i)i linearization 15 1 5 14.66 14.66-1 1 2 3 1 2 3 1 2 3 injected dc-current, I injected dc-current, I injected dc-current, I numerical ω(i) c(i) membrane voltage, (mv) amplitude numerical theoretical frequency, 2π/period (rads/ms) 2.5 2 1.5 1.5 frequency theoretical numerical Figure 6.12: Supercritical Andronov-Hopf bifurcation in the I Na,p +I K -model with lowthreshold K + current (see Fig.6.11). Dots represent numerical simulation of the full model, continuous curves represent analytical results using the topological normal form (6.8, 6.9). Example: The I Na,p +I K -Model

-2 Subcritical Andronov-Hopf bifurcation max/min of oscillations of membrane potential, mv -4 u nstable stable limit cycles unstable -6-8 45 5 55 injected current, I I=48 I=47 176 Bifurcations I=45 unstable limit cycles n V I=49 stable unstable I I=48.75 subcritical Andronov-Hopf bifurcation at I=48.8 max/min of oscillations of membrane potential, mv -2-4 u nstable limit cycles stable unstable -6-8 45 5 55 injected current, I I=47 I=48 I=43 1.8.6.4 V-nullcline imaginary part, ω C eigenvalues c + iω stable unstable real part, c.2 I=45 n-nullcline -8-6 -4-2 2 membrane voltage, V (mv) I=43 1.8.6 V-nullcline imaginary part, ω subcritical Andronov-Hopf bifurcation at I=48.8 C eigenvalues c + iω Figure 6.16: Subcritical Andronov-Hopf bifurcation in the INa,p+IK-model. As the b furcation parameter I increases, an unstable limit cycle (dashed circle; see also Fig.6.1 shrinks to an equilibrium and makes it lose stability. Parameters are as in Fig.4.1 except gl =1,gNa = gk = 4, and the Na + activation function has V1/2 = 3 mv an k =7..4 stable unstable

Bistability in the I Na,p + I K model To be explained later 174 1 K+ activation variable, n.5 stable limit cycle unstable limit cycle n-nullcline stable equilibrium (rest) -8-7 -6-5 -4-3 -2-1 1 membrane voltage, V (mv) V-nullcline Figure the I Na, limit c often su (solid c neurona

Delayed loss of stability To be analyzed later (also)... Bifurcations 177 membrane potential, V (mv) -2-4 -6 (a) stable limit cycle unstable limit cycle stable subcritical Andronov-Hopf bifurcation unstable delayed loss of stability membrane potential, V (mv) -2-4 -6 (b) stable limit cycle unstable limit cycle noise-induced sustained oscillations subcritical Andronov-Hopf bifurcation injected current, I(t) 6 4 2 4 6 8 1 12 14 16 18 2 time, t

Outline 1 Codim 1 bifurcations of equilibria 2 Codim 1 bifurcations of limit cycles 3 Additional bifurcations examples (3d) 4 Codim 2 bifurcation: Bogdanov-Takens 5 Excitability 6 Integrators / Resonators 7 Summary Transitions from spiking to resting. (SNIC, superah,sn-lc,saddle homoclinic)

drastically increasing interspike intervals as the system approaches the bifurcation sta Neural (see Fig.6.19). properties of bifurcations In contrast, supercritical Andronov-Hopf bifurcation results in oscillations wi vanishing amplitude, as one can clearly see in Fig.6.19. If neither the frequency n the amplitude vanishes, then the bifurcation is of the fold limit cycle type. Indee Bifurcation of a limit cycle attractor Amplitude Frequency saddle-node on invariant circle nonzero A I I b supercritical Andronov-Hopf A I I b nonzero fold limit cycle nonzero nonzero saddle homoclinic orbit nonzero A ln I I b Figure 6.18: Summary of codimension-1 bifurcations of a limit cycle attractor on plane. I denotes the amplitude of the injected current, I b is the bifurcation value, an A is a parameter that depends on the biophysical details.

6.2.2 Supercritical Andronov-Hopf Saddle node of limit cycle bifurcation 1/2 A stable limit cycle can shrink to a point via supercritical Andronov-Hopf bifurcation in Fig.6.21, which we considered in section 6.1.3. Indeed, as the bifurcation parameter changes, e.g., the injected DC current I in Fig.6.11 decreases, the amplitude of the limit cycle attractor vanishes, and the cycle becomes a stable equilibrium. As we showed in section 6.1.3 (see Fig.6.12), the amplitude scales as I I b when the bifurcation parameter approaches the bifurcation value I b. stable unstable limit cycle limit cycle fold limit cycle Figure 6.22: Fold limit cycle bifurcation: a stable and an unstable limit cycle approach and annihilate each other. 6.2.3 Fold Limit Cycle A stable limit cycle can appear (or disappear) via the fold limit cycle bifurcation depicted in Fig.6.22. Consider the figure from left to right, which corresponds to the

Saddle node of limit cycle bifurcation 2/2 I=42.5 stable unstable limit cycles fold limit cycle I=42.35 Bifurcations 183 I=43 stable limit cycles fold limit cycle I=42.18 I=42.5 unstable limit cycles 1 I=42 fold limit cycle I=42.35 min/max of oscillation of membrane potential, mv -2 bifurcation -4-6 stable limit cycles unstable limit cycles stable equilibria.5 V-nullcline n-nullcline -8 4 45 5 injected current, I -8-6 -4-2 membrane voltage, mv fold limit cycle I=42.18 Figure 6.23: Fold limit cycle bifurcation in the INa,p+IK-model. As the bifurcation parameter I decreases, the stable and unstable limit cycles approach and annihilate each other. Parameters are as in Fig.6.16. I=42 1 llation of tial, mv -2 stable limit cycles V-nullcline n-nullcline

homoclinic orbit homoclinic orbit Saddle-Node homolinic bifurcation 1/4 Andronov-Leontovich Figure 6.24: Bifurcation 1939 diagram of the I Na,p +I K -model. Parameters are as in Fig.6.16. Typically W u W s =. 4 41 42 43 44 45 46 47 48 49 5 injected dc-current, I stable limit cycle saddle saddle saddle a. supercritical saddle homoclinic orbit bifurcation unstable limit cycle saddle saddle saddle b. subcritical saddle homoclinic orbit bifurcation

.6 Saddle-Node homolinic bifurcation 2/4.8.4 V-nullcline stable limit cycle n-nullcline I=7 limit cycles n V I.2-8 -6-4 -2 membrane voltage, mv I=4 node saddle Bifurcations 187 Estimate of period: T ln(µ µ c ) node stable limit cycle saddle stable limit cycle I=3.8.8.6.4 V-nullcline stable limit cycle n-nullcline I=7 limit cycles n V I homoclinic orbit homoclinic orbit.2 node saddle I=1-8 -6-4 -2 membrane voltage, mv stable limit cycle node saddle stable limit cycle I=4 I=3.8 node saddle min/max of oscillations of membrane potential, mv -2-4 -6 homoclinic bifurcation saddles nodes limit cycles -8 2 4 6 injected dc-current, I node saddle homoclinic orbit homoclinic orbit Figure 6.28: Saddle homoclinic orbit bifurcation in theina,p+ik-model with parameters as in Fig.4.1a and fast K + current (τ(v )=.16). As the bifurcation parameter I decreases, the stable limit cycle becomes a homoclinic orbit to a saddle. node saddle I=1 min/max of oscillations of membrane potential, mv -2-4 -6 homoclinic bifurcation saddles nodes limit cycles -8 2 4 6 node saddle

Saddle-Node homolinic bifurcation 3/4 188 Bifurcations.8-1.4.7.2.6 T2-2 K + activation gate, n.5.4.3.2 V-nullcline T1 saddle -6-55 -5 n-nullcline limit cycle membrane voltage, V (mv) -3-4.1 T1 T2-8 -7-6 -5-4 -3-2 -1 1 membrane voltage, V (mv) -5 T1 T2-6 1 2 3 4 5 6 time, ms Figure 6.29: The period of the limit cycle is T = T1 + T2 with T2 as the cycle approaches the saddle equilibrium. Shown is the INa,p+IK-model with I = 3.5. (dashed part of the limit cycle in the figure) and T2 denote the time spent in the small neighborhood of the saddle equilibrium (continuous part of the limit cycle in the shadowed region), so that the period of the limit cycle is T = T1 + T2. While T1 is relatively constant, T2 as I approaches the bifurcation value Ib =3.8, and the limit cycle approaches the saddle. In exercise 11 we show that T2 = 1 ln{τ(i Ib)}, λ1 where λ1 is the positive (unstable) eigenvalue of the saddle, and τ is a parameter that depends on the size of the neighborhood, global features of the vector field, and so on. We can represent the period, T, in the form

6.3: Frequency of spiking in the I Na,p +I K -model with parameters 8 near a saddle homoclinic orbit bifurcation. Dots are numerical results ous curve is ω(i) =1λ(I)/{ ln(.2(i 3.814))}, where the eigen Saddle-Node homolinic bifurcation 4/4 tions frequency (Hz) 4 3 2 2 numerical theoretical 1 1/ln 1 3.8 3.9 3 3.1 3.2 3.3 3.4 3.5 3.6 injected dc-current, I

Subcritical Flip bifurcation Period doubling Saddle-Focus Homoclinic Orbit Bifurcation Subcritical Flip Bifurcation

Neimark-Sacker bifurcation Invariant torus creation Subcritical Flip Bifurcation Subcritical Neimark-Sacker Bifurcation

Bogdanov-Takens bifurcation Theorem Consider the DS ẋ = f (x; µ) R 2, f smooth, f (; ) = with two zeros eigenvalues λ 1,2 =. If A() f x (, ), a 2 () + b 11 (), b 2 the map (x, µ) ( f (x, µ), tr ( ) ( f x, det f )) x is smooth in (, ), then the DS is locally topologically equivalent to the DS η 1 = η 2 η 2 = β 1 + β 2 η 1 + η1 2 + sη where s = sign a 2() + b 11 () 1η 2 b 2 () Truncation does not change the dynamics.

BT bifurcation diagram 196 Bifurcations 1 2 b 1 2 SN2 3 AH 4 1 BT 4 a 3 AH 4 SHO AH 2 3 SN1 SN 1 SN 2 SN 1 SN 2 SHO BT supercritical, s< subcritical, s> Figure 6.38: Bogdanov-Takens (BT) bifurcation diagram of the topological normal form (6.11). Abbreviations: AH, Andronov-Hopf bifurcation; SN, saddle-node bifurcation; SHO BT

bifurcation (at some I 3) occurs, which is subcritical, giving birth to an unstable limit Example cycle. of athesubcritical phase portrait at I 3 is bifurcation locally topologically equivalent to the one marked SHO in Fig.6.38. Similarly, the phase portrait at I 4 is locally equivalent to the one labeled 2 in Fig.6.38. The unstable cycle shrinks to the equilibrium and makes it lose stability via a subcritical Andronov-Hopf bifurcation at some I 5, which corresponds to case AH in Fig.6.38. Further increase of I converts the unstable focus into an unstable node, which approaches the saddle and disappears via the saddle-node bifurcation SN 1 in Fig.6.38 (not shown in Fig.6.4). I = I 1 I 2 big homoclinic orbit I 3 I 4 I 5 small homoclinic orbit subcritical Andronov-Hopf bifurcation Figure 6.4: Transformations of phase portraits of a neuronal model near the subcritical Bogdanov-Takens bifurcation point as the magnitude of the injected current I increases (here I k+1 >I k). Shaded regions are the attraction domains of the equilibrium corresponding to the resting state.

Neuronal Excitability Romain Veltz / Olivier Faugeras October 15th 214 ENS - Master MVA / Paris 6 - Master Maths-Bio (213-214)

Hodgkin s Classification Class I (Figures from Izhikevich) Excitability 219 pa I (pa) Class 1 excitability pa Layer 5 pyramidal cell Brainstem mesv cell I=32 I=8 I=6 I=4 I=2 pa I (pa) 1 ms 4 mv I=1 I=6 I=5 1 2 3 injected dc-current, I (pa) 2 ms 2 mv I=4Figure 7.3: Top: Typical responses of membrane p Action potentials can be DC current of various magnitudes I. Bottom: Cor I=2relations generated are qualitatively with arbitrarily different. Shown loware reco fromfrequency, rat primary visual depending cortex (left) onand themesv neu asymptotic frequency is 1/T, where T is ta last I (pa) strength two spikes inofa long the spike applied train. current. pa asymptotic firing frequency, Hz 4 3 2 1 F-I curve asymptotic firing frequency, Hz 25 2 15 1 5

Hodgkin s Classification: Class I Excitability 217 excitable bistable oscillatory saddle-node on invariant circle bifurcation saddle-homoclinic orbit bifurcation saddlenode bifurcation

I (pa) Hodgkin s Classification pa Class II (Figures from Izhikevich) 219 Class 1 excitability pa I (pa) Class 2 excitability Brainstem mesv cell I=1 I=6 I=5 asymptotic firing frequency, Hz 4 3 2 1 F-I curve 1 2 3 injected dc-current, I (pa) asymptotic firing frequency, Hz 25 2 15 1 5 F-I curve 5 1 15 injected dc-current, I (pa) V I=4 I=2 pa 2 ms Figure 7.3: Top: Typical 2 mvresponses of membrane Action potentials of two neurons are generated to steps of DC current of various magnitudes I. Bottom: Corresponding frequency-current (F-I) relations are qualitatively different. Shown are in arecordings certainof frequency layer 5 pyramidal bandneurons that from rat primary visual cortex (left) and mesv is relatively neuron frominsensitive rat brainstem to (right). The asymptotic frequency is 1/T, where T is taken to be the interval between the last two spikes in a long spike train. I (pa) changes in the strength of the applied current.

orbit bifurcation Hodgkin s Classification: Class II node bifurcation supercritical Andronov-Hopf bifurcation fold limit cycle bifurcation subcritical Andronov- Hopf bifurcation Figure 7.2: Excitable dynamical systems bifurcate into oscillatory ones either directly or indirectly, via bistable systems.

Hodgkin s Classification 1 pa Class III (Figures from Izhikevich) pa 5 pa -6 mv Figure 7.4: Class 3 excitability of a mesv neuron of rat brainstem (contrast with Fig.7.3). 2 mv 1 ms -2mV 5 ms 3 mv subthreshold oscillations 7 pa 1 pa Figure 7.5: Class 3 excitability of a layer 5 pyramidal neuron of rat visual cortex. The inset shows subthreshold oscillations of membrane potential. L5 Pyramidal neuron in rat visual cortex. One spike is generated in Instead, they can act as threshold elements reporting when the strength of an input is response aboveto a certain a current value. Both step. properties Repetitive are important(tonic) in neural computations. spiking can be Hodgkin also observed that axons left in oil or seawater for long periods exhibited generated only for extremely strong injected currents or not at all. Class 3 neural excitability. A single action potential is generated in response to a pulse of current. Repetitive (tonic) spiking can be generated only for extremely

Hodgkin s Classification Class III (Figures from Izhikevich) Class 3 neural excitability occurs when the resting state remains stable for any fixed I in a biophysically relevant range Excitability 223.25.2.15 w.1.5 w-nullcline I=.3 V-nullcline I= I=.6.45.3.15.1.4.2.2.4.6.8 V Figure 7.8: Class 3 excitability in FitzHugh-Nagumo model (4.11, 4.12) with a =.1,b=.1,c=. The model fires a single spike for any pulse of current. Example of the Fitzhugh-Nagumo model V = V (a V )(V 1) w + I, ẇ = bv cw, a =.1, b =.1 to the slow ramp. In contrast, a quick membrane depolarization due to a strong step +

Ramps, Steps, and Shocks 224 Excitability.2.15.1.5 I 1 I 1 I 1 K + gating variable, n I I -7-6 -5-4 -7-6 -5-4 -7-6 -5-4 membrane potential, V (mv) membrane potential, V (mv) membrane potential, V (mv) -2-4 -6-8 V(t) I(t) I 1 I 1 I 1 I I slow ramp from I to I 1 step from I to I 1 shock pulse at I 1 (a) (b) (c)

ure 7.1: The INa,p+IK-model undergoes subcritical Andronov-Hopf bifurcation, yet exhibit low-frequency firing when pulses (but not ramps) of current are injected. ameters: C = 1, I =, EL = 66.2, gl = 2, gna = 5, gk = 4.5, m (V ) has 2 = 3 and k =1,n (V ) has V1/2 = 34 and k =13,andτ(V )=1,ENa =6 and EK = 9 mv. The shaded region is the attraction domain of the resting te. The inset shows a distorted drawing of the phase portrait. The I Na,p + I K model, Class I or II? citability 225 (a) (b) frequency K+ gating variable, n pulses ramp frequency-current (F-I) relation 3.88 subcritical Andronov-Hopf bifurcation (d) I=5.25.7 near subcritical Andronov-Hopf bifurcation Andronov-Hopf homoclinic 5.25 current, I (c) (e).7 K+ gating variable, n I=3.8866 near saddle homoclinic orbit bifurcation Class II for ramps, I for Steps subcr. AH at I = 5.25 for ramps SN homoclinic for I 3.89 for steps near BT.1-8 membrane potential, V (mv).1-8 membrane potential, V (mv)

Bistability 226 Excitability saddle-node bifurcation subcritical Andronov-Hopf bifurcation inhibitory pulse attraction domain n-nullcline spiking limit cycle excitatory pulse n V-nullcline V V-nullcline exc. pulse spiking limit cycle unstable n-nullcline inhibitory pulse attraction domain of spiking limit cycle Figure 7.11: Coexistence of stable equilibrium and spiking limit cycle attractor in theexcitatory I Na,p +I K -model. / inhibitory Left: pulses The resting can shift state the is neuron about tofrom disappear its resting via saddle-node state to bifurcation. repetitiveright: firing. The resting state is about to lose stability via subcritical Andronov- Hopf bifurcation. Right (left) arrows denote the location and the direction of an

7.2.2 Frequency Preference and Resonance Integrators vs. Resonators 1/2 A standard experimental procedure to test the propensity of a neuron to subthreshold oscillations is to stimulate it with a sinusoidal current having slowly increasing frequency (called a zap current), as in Fig.7.17. The amplitude of the evoked oscillations of the membrane potential, normalized by the amplitude of the stimulating oscilla- -4 mv 1 Hz membrane potential response, mv pa 5 mv 1 sec 2 pa zap current, pa 14 Hz spikes cut 2 Hz amplitude of the response integrators resonators resonance frequency of stimulation Figure 7.17: Response of the mesv neuron to injected zap current sweeping through a range of frequencies. Integrators and resonators have different responses. 1 Presence of sub-threshold oscillations 2 Exp. mechanism to test their presence

Integrators vs. Resonators 2/2 Excitability 233-42 integrators -42-42 -42-42 -42 potential (mv) potential (mv) potential (mv) potential (mv) potential (mv) potential (mv) -52 5 ms -52 1 ms -52 15 ms 1 2 3 time (ms) 1 2 3 time (ms) 1 2 3 time (ms) resonators -52 5 ms -52 1 ms -52 15 ms 1 2 3 time (ms) 1 2 3 time (ms) 1 2 3 time (ms) Figure 7.18: Responses of integrators (top) and resonators (bottom) to input pulses having various inter-pulse periods. Integrators prefer high frequencies: detect coincidences. Resonators are selective in a frequency band.

natural period, which is 1 ms in Fig.7.18 and Fig.7.2, the second pulse arrives during the Selective rising phase response: of oscillation, excitation and increases orthe inhibition amplitude of oscillation even further. In this case the effects of the pulses add up. The third pulse increases the amplitude of oscillation even further, thereby increasing the probability of an action potential, as in Fig.7.2. 5ms 1ms 15ms non-resonant burst resonant burst non-resonant burst Figure 7.2: Experimental observations of selective response to a resonant (1 ms Mesencephalic V neurons in brainstem having subthreshold membrane interspike period) burst in mesencephalic V neurons in brainstem having subthreshold membrane oscillations oscillations with a natural with a period naturalaround period around 9 ms Three 9 ms; consecutive see also Fig.7.18. voltage Three consecutive traces are voltage shown to traces demonstrate are shownsome to demonstrate variability some of thevariability result of the result. (Modified from Izhikevich et al. 23).

Selective response: excitation or inhibition Excitability 235 inhibitory input 5ms 15ms 1ms non-resonant burst non-resonant burst resonant burst Figure 7.21: Experimental observations of selective response to inhibitory resonant burst Experimental in mesencephalic observations V neurons of selective in brainstem response havingtooscillatory inhibitorypotentials resonantwith burst the natural in mesencephalic period around V9neurons ms. (Modified in brainstem from Izhikevich having oscillatory et al. 23). potentials with the natural period around 9 ms. (Modified from Izhikevich et al. 23). If the interval between pulses is near half the natural period, e.g., 5 ms in Fig.7.18

leads to a decrease in oscillation amplitude. The spikes effectively cancel each other in Selective this case. response: Similarly, the spikes excitation cancel each other or inhibition when the interpulse period is 15 ms, which is 6 percent greater than the natural period. The same phenomenon occurs Geometrical explanation for inhibitory synapses, as we illustrate in Fig.7.21. Here the second pulse increases (decreases) the amplitude of oscillation if it arrives during the falling (rising) phase. We study the mechanism of such frequency preference in exercise 4, and present its geometrical illustration in Fig.7.22. There, we depict a projection of the phase portrait spike excitatory pulses spike inhibitory pulses spike small PSP Hodgkin-Huxley model n+m+h spike V resonant non-resonant 1 2 1 2 1 2 2 1 2 1 Figure 7.22: (Left) Projection of trajectories of the Hodgkin-Huxley model on a plane. (Right) Phase portrait and typical trajectories during resonant and non-resonant response of the model to excitatory and inhibitory doublets of spikes. (Modified from Izhikevich 2a). Important temporal coherence of pulses.

Influence of synaptic bombardment. Frozen 238noise experiment Excitability (g) 9 ms (f) (e) (d) (c) (b) (a) membrane potential (mv) spike raster injected current (pa) 7 ms 6 ms 4 ms no burst input 2-2 -4-6 2 frozen noise burst input 2 mv 6.7 ms 5 1 15 2 25 3 35 4 45 5 time (ms)

Threshold definition Spike th. definiton not obvious Excitability Excitability 1ms 239 1ms 1 mv threshold? 1 mv threshold? Figure 7.25: Finding the threshold in the Ho Huxley model. Figure 7.25: Finding the threshold in the Hodgkin- Huxley model. squid axon model squid axon mv model mv mv -5 mv mv -5 mv -5 mv 5 ms -5 mv 5 ms Figure 5 ms7.26: Variable-size action potentials 5 inmssquid giant axon and revised Hod Huxley model (Clay 1998) in response to brief steps of currents of variable magni Figure 7.26: Variable-size (Data actionprovided potentials byinjohn squidclay.) giant axon and revised Hodgkin- Huxley model (Clay 1998) in response to brief steps of currents of variable magnitudes. (Data provided by John Clay.)

Threshold manifolds for the I Na,p + I K model Integrators (close to saddle-node bifurcation) have a threshold manifolds, given by stable manifold of fold point. For resonators, it depends on bifurcation type Bistable regime: (close to sub. AH): the unstable LC is a th. manifold. In all other cases, there is no well defined th. manifold.

Threshold manifolds for the I Na,p + I K model 24 Excitability K + activation K + activation.6 (a) integrator threshold manifold -7-6 -5-4 small EPSP small EPSP (c) resonator half-amplitude spike spike spike.5.2 1.8.6.4 (b) resonator threshold manifold small EPSP -7-6 -5-4 (d) resonator canard trajectory P spike spikes.1 threshold set.2-6 -5-4 -3 membrane potential, mv -7-6 -5-4 -3-2 membrane potential, mv

Spike emission: excitation or inhibition? 244 Excitability (a) integrator (b) resonator K + activation gate inhibition excitation -6-4 -2 membrane potential, mv inhibition excitation -6-4 -2 membrane potential, mv Figure 7.3: Direction of excitatory and inhibitory input in integrators (a) and resonators integrator (b). cannot spike in response to a hyper-polarizing current, a An resonator can.

-6-4 -2 membrane potential, mv Spike emission: excitation or inhibition? -6-4 -2 membrane potential, mv Figure 7.3: Direction of excitatory and inhibitory input in integrators (a) and resonators (b). K + activation gate 1 mv 1 ms inhibitory pulse excitatory pulse inhibitory pulse excitatory pulse -6-4 -2 membrane potential, mv Figure 7.31: Postinhibitory facilitation: A subthreshold excitatory pulse can become An hyper-polarizing current (inhib.) can enhance the effect of subsequent superthreshold if it is preceded by an inhibitory pulse. excitatory pulses. 7.2.8 Inhibition-Induced Spiking

Latency to first spike 246 2 mv -75 mv 5 ms pa 2 pa Figure 7.3 and thres layer 5 ne vitro of ra To understand the dynamic mechanism of such an inhibition-in Property of integrators need to consider the geometry of the nullclines of the model, de Allow coding(right). of input Note strengh how the in latency positiontooffirst thespike V -nullcline depends on I. Ne Less sensitive nullcline to noise, down since andonly leftward prolonged so that inputs the vertex can cause of itsspikes left knee, marke to the left. As a result, the equilibrium of the system, which is the V - and h-nullclines, moves toward the middle branch of the cubic V

Figure 7.35: Bifurcation mechanism of latency to first spike when the injected DC current steps from I =toi>i b, where I b is a bifurcation value. The shaded circle denotes the region where vector field is small. Phase portraits of the I Na,p +I K -model Latency to first spike Excitability 247.1 saddle-node bifurcation V-nullcline when I>I b V-nullcline when I= n-nullcline K + activation gate 1.5 Andronov-Hopf bifurcation n-nullcline I= V-nullcline when I>I b V-nullcline when -7-6 -5-4 membrane potential, mv -7-6 -5-4 -3 membrane potential, mv

Flipping from an Integrator to a Resonator Mitral cells can be quickly switched from being integrators to being resonators by synaptic input. For resonators, it depends on bifurcation type high-states are attractive focus down-states are stable node

Integrator Resonator 1/2 248 Excitability 1 ms 1 mv oscillations 2 ms 2 mv -6 mv up-state down-state integrator resonator n-nullcline V-nullcline

Figure 7.36: Bistability of the up-state and down-state of mitral cells in rat main olfactory bulb. The cells are integrators in the down-state and resonators in the upstate. Membrane potential recordings are modified from Heyward et al. (21). The shaded area denotes the attraction domain of the up-state. Integrator Resonator 2/2 up-state -6 mv down-state integrator resonator n-nullcline V-nullcline up-state (focus) down-state (node)

BT coming into play? -6 mv 5 mv 1 ms Figure 7.37: Fast subthreshold oscillations during complex spikes of cerebellar Purkinje neuron of a guinea-pig. (Data was by Yonatan Loewenstein.) Need to combine in 2d a SN with a AH: integrator (saddle-node bifurcation)? resonator (Andronov-Hopf bifurcation) V-nullcline n-nullcline? Figure 7.38: Is there an intermediate mode between integrators and resonators? A similar phenomenon was observed in a cerebellar Purkinje neuron (see Fig.7.37). It acts as an integrator in the down-state, but has fast (> 1 Hz) subthreshold oscillations in the up-state, and hence can act as a resonator.

Bogdanov-Takens 1/3 In the 25I Na,p + I K model (parameters E L, V 1/2 ), the nullclines are Excitability parallel close to the left knee of the fast nullcline ate, n K + activation gate, n.5.4.3.2.1-6 -4-2.5 n-nullcline n-nullcline Bogdanov-Takens bifurcation V-nullcline V-nullclin te, n K + activation gate, n integrator (near saddle-node bifurcation).5.4.3.2.1-65 -6-55 -5.5.4

te, n K + activation gate, n K + activation gate,.4 Bogdanov-Takens 2/3.3.2.1 A small change -6in the -4 parameter -2 V 1/2 can give a Saddle-Node: -65-6 -55i.e. an -5 integrator.5.4.3.2.1-6 -4-2.6.5 n- cline V-nullcline V-nullclin n-nullcline n-nullcline e, n K + activation gate, n K + activation gate, n.3.2.1 integrator (near saddle-node bifurcation) resonator (near Andronov-Hopf bifurcation).5.4.3.2.1 threshold -65-6 -55-5.5.4

K + activation gate, n K + activatio.3 Bogdanov-Takens 3/3.2.1-6 -4-2 or a resonator via a stable focus..6.5.4.3.2.1 n-nullcline K + activation gate, n K + activation resonator (near Andronov-Hopf bifurcation) V-nullcline.3.2.1 threshold -65-6 -55-5.5.4.3.2.1-6 -4-2 membrane potential, V (mv) -65-6 -55-5 membrane potential, V (mv) Figure 7.39: Bogdanov-Takens bifurcation in the I Na +I K -model (4.1, 4.2). Parameters are as in Fig.4.1a, except n (V ) has k =7mVandV 1/2 = 31.64 mv, E leak = 79.42 and I =5. Integrator: V 1/2 = 31 mv and I =4.3. Resonator: V 1/2 = 34 mv and I =7.

Summary 1/2 determine the type of bifurcation of the resting state, a For example, a bistable integrator corresponds to a sadd monostable resonator corresponds to a supercritical Andr grators and resonators have drastically different neurocom marized in Fig.7.15 and discussed next (the I-V curves ar coexistence of resting and spiking states YES (bistable) NO (monostable) subthreshold oscillations NO (integrator) YES (resonator) saddle-node subcritical Andronov-Hopf saddle-node on invariant circle supercritical Andronov-Hopf Figure 7.14: Cla monostable/bistabl according to the bif

Summary 2/2 23 Excitability properties integrators resonators bifurcation saddle-node on invariant circle saddle-node subcritical Andronov-Hopf supercritical Andronov-Hopf excitability class 1 class 2 class 2 class 2 oscillatory potentials frequency preference no no yes yes I-V relation at rest non-monotone monotone spike latency large small threshold and rheobase well-defined may not be defined all-or-none action potentials yes no co-existence of resting and spiking post-inhibitory spike or facilitation (brief stimuli) inhibition-induced spiking no yes yes no no yes no possible