Stable

Eq. M

anif

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SN Line

uo

T(uo)

v

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Unstable Eq. Manifold

Multirhythmic Bursting

General Mechanism for Multistability in

Jon Newman and Robert Butera Laboratory for Neuroengineering, Georgia Institute of Technology, Atlanta, GAMultistable DynamicsMultistability occurs when a system’s intrinsic dynamics support multiple stable modes of operation. For example, consider a ball rolling on a hilly surface. There are many ‘basins’ for the ball to roll into and therefore multiple locally stable points in the state space of the dyanmical system that describes the ball’s motion.

Multistability is a widely hypothesized mechanism for the storage of memory in neural systems [1, 2, 3, 4]. If a neural system is multistable, sensory input may serve to perturb the state of a neural circuit’s activity into and out of the basins of attraction of various stable modes. The neural system thus acts as a switch in response to external input, with a memory instantiated by its activity state.

We present a general mechanism for a multistable dynamics in single, ‘parabolic’ bursting neurons that does not rely on a delayed feedback. We provide experimental evidence for its existence in neurons of the aquatic mullusc Aplysia Californica. These preliminary experimental results indicate that single neurons may be capable of dynamically storing information for longer time scales than typically attributed to non-synaptic mechanisms.

Parabolic Bursting is a Balance

Reduction and Generalization • CCB system can be reduced to a 1-dimensional return map that switches be-tween two modes, P(u).

- Can be verified for complicated biophysical models [2, 3]

• P(u) can account for all behaviors seen in single cell parabolic bursting stud-ies, both modeling and experimental [2, 3, 5] (fig. 5b and fig. 6).

- Map switches between two moduli, s1 and s2, which act over sets of size r1 and r2, respectively- Fixed point at the origin is a nominal bursting solution with N spikes- Constrain|s2| > 1 so that multiple basins of attraction can exist- Average slope, save = |(s1r1+s2r2)/(r1+r2)| < 1, guarantees the existence of at least one attractor

1

0 1-1

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S1

M M

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u

uNB NB

NB

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u

u u

s1 s 2 Dynamics

– |s2 |< 1 s2 must have modulus greater than 1|s1 |< 1 s2 < −1 Non-biological because s ave < 0s1 > 1 s2 > 1 Non-biological because s ave > 1 indicating that the system

is repeller0 < s 1 < 1 s2 < −1 Map is not well defined

s1 < −1 s2 > 1 Chaotic and possibly multirhythmic since the full map isformed by a chain of sawtooth maps with only unstablefixed points

−1 < s 1 < 0 s2 > 1 Attractors are oscillatory fixed points and system ispossibly multirhythmic

0 < s 1 < 1 s2 > 1 Attractors are non-oscillatory fixed points and system ispossibly multirhythmic

s1 > 1 s2 < −1 Chaotic and possibly multirhythmic since the full map isformed by a chain of tent maps with only unstable fixedpoints

Table 1 The qualitative behavior of the piecewise linear maps shown in fig. 5 is described for each parameter regime. The bottom section describes relevant parameter ranges that can fully account for the dynamics of prior multirhythmic bursting models.

Fig 6 P(u) under the parameter sets outlined in table 1. Qualitative dy-namics are classified by the follow-ing symbols: M - may support multi-rhythmicity, C - chaotic, and NB - non-biological. Maroon sections of the maps show a single link in a chain of chaotic maps.

Conclusions• Since time scales associated with short term memory are smaller than those of morphological plasticity it is possible that neural system employ multistable dynamics as a memory

• We have shown a general mechanism for multistable bursting that does not re-quire a delayed feedback and is supported by a large range of parameters

• Even if the CCB system is not truly multistable, the shape of P(u) indicates that the system will always have a non-monotonicity in the contraction of the vector field about the attractor (in C ) which may allow temporal amplification of synaptic perturbations

Fig 2 CCB solution (dark line, ) alter-nates between spiking and silent re-gimes separated by a line of saddle note on invariant circle bifurcations. When silent, v is confined to its stable equilibrium manifold allowing the use of a 1-dimensional section () to derive a Poincaré return map.

• Many neurons burst paraboli-cally – the spike period profile looks like a parabola.

• Parabolic bursters actually have root scaling in firing rate (fig. 1).

• Parabolic bursters are well de-scribed by ‘circle/circle’ bursting (CCB) systems [5]

• Two slow variables drive the fast sub-system across saddle-node on invariant circle bifurcation (SNIC) (fig. 2).

• Dynamically, CCB solutions are a balance between expansive spik-ing events on, and intrinsic dissi-pation of, the slow subsystem.

Model and Mechanism• Multirhythmic bursting (MRB) occurs when different numbers of spiking events are capable of balancing slow contrac-tion (fig. 3a,b).

• Three contraction metrics for spiking (Cs), silent (Cr), and combined (C) re-gions of state space by measuring the distance covered along the Poincaré section by trajectories of eq. (1).

• A necessary condition for the existence of a multirhythmic CCB system is it must support a non-monotonic C = Cs ˉ Cr that crosses 0 (fig. 3c).

• The non-monotonicity of C is caused by spike addition.

• New bursting solutions are formed by saddle node bifurcations of closed orbits (SCO’s) that are equivalent to saddle node bifurcations of C.

In Biological Neurons

• We searched for MRB in parabolic bursters of Aplysia’s abdominal ganglion.

• Hyperpolarizing current perturbations and subsequent rebound volleys were used to induce spike addition.

• Found indications of attractor switching in response to these pertur-bations (fig. 4)

(1) Consistent increase or decrease in burst period (figs. 3a and 4c)

(2) Change in the number of spikes in the active phase (fig. 4b)

• ‘Fading multirhythmicity’ was seen quite often (fig. 5)

- Cell not truly mulitrhythmic, but close to an SCO

- Dynamics near the ruins of the SCO is slow leading to temporal amplification of a perturbation

References[1] J. J. Hopfield. Proc Natl Acad Sci U S A, 79(8):2554, 1982.[2] C. C. Canavier, D. A. Baxter, J. W. Clark, and J. H. Byrne. J Neurophysiol, 69(6):2252, 1993.[3] R. J. Butera. Chaos, 8(1):274, 1998.[4] H.A. Lechner, D.A. Baxter, J.W. Clark, and J.H. Byrne. Bistability and its regulation by serotonin in the endogenously bursting neuron R15 in Aplysia. J Neurophysiol 1996;75:957-62.[5] E.M. Izhikevich. International Journal of Bifurcation and Chaos, 10:1171, 2000.

AcknowledgementsJ. Newman was supported by Georgia Tech's NSF IGERT program on Hybrid Neural Microsystems (DGE-0333411) and an NSF Graduate Research Fellowship. Experimental and modeling work was also supported by grants from the NSF (CBET-0348338) and NIH (R01-HL088886), respectively, to R. Butera.

321.9C73

Fig 3 (a) Voltage traces of three coexisting CCB solutions of eq. (1). Note that each limit cycle has a distinct period. (b) The three solutions projected into the plane of the slow variables. Contraction metrics C, Cs, and Cr are illustrated on an example trajectory. (c) Cr, Cs, and C for the system de-fined by eq. (1). The three stable solution occur when C0.

u 2

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_

(1a)(1b)(1c)

v(t) = vc{vr ← v(t), u1(t) + d1 ← u1(t), u2(t) + d2 ← u2(t)}

Fig 1 Recording from identified bursting neuron R15 of Aplysia’s abdominal ganglion. Voltage trace shows a single burst period. The corresponding spike fre-quency profile (open circles) of the burst is compared to root scaling (grey lines).

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Fig 4 Current perturbation experiment showing multirhythmic bursting in R15. (a) Voltage trace and current perturbation. (b) Spikes per burst and example pre/post-perturbation active phases. (c) Low pass filtered voltage trace before (blk.) and after (org.) showing a constant phase advance as predicted by the model.

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Fig 5 Fading multirhythmic bursting in L3. (a) Voltage trace and current perturbation. (b) Hypothesized equivalent return map close to a SCO. (c) Perturbation and subsequent intermittent dynamics of the map.

Single Bursting Neurons