Transcript Slide 1

Phase Bursting Rhythms in Inhibitory Rings
Matthew Brooks, Robert Clewley, and Andrey Shilnikov
Abstract
Strong Coupling: Symmetric and Asymmetric Motifs
A multifunctional central pattern generator (CPG) is able to produce bursting polyrhythms that
determine locomotive activity in an animal: for example, swimming and crawling in a leech. Each
rhythm corresponds to a periodic or aperiodic attractor of the CPG. We study the multistability (stable
coexistence) of these attractors, as well as the switching between them, using a model of a
multifunctional CPG. We consider a Hodgkin-Huxley type model of a leech heart interneuron, three of
which are mutually coupled in a ring by fast inhibitory synapses. Each neuron is a 3D system of
deterministic ODEs exhibiting periodic bursting, where a burst consists of episodes of fast tonic spiking
and slow quiescence.
• Burst rhythm outcomes are computed for
discretized values of phase pairs (Ф1, Ф2) with Ф1, Ф2
in [0,1] . Shown above is a symmetric strongly
coupled case, with gij = 0.1, for all i,j.
• As the phases are varied with respect to Tiso, the
resulting burst rhythm shifts; one cell in the motif is
always anti-phase with the other two.
We employ the tools of dynamical systems and bifurcation theory to understand the rhythmic outcomes
of the network. We show that the problem can be effectively reduced to the phase plane for the phase
differences of the neurons on the bursting periodic orbit. Using computer assisted analysis, we examine
the bifurcations of attractors and their basins in the phase plane, separated by repellers and separatrices
of saddles which are the hidden organizing centers of the system. These structures determine the
resulting bursting rhythms produced globally by the CPG. By varying the coupling synaptic strength, we
examine the emerging dynamics and properties synchronization patterns produced by symmetric and
asymmetric CPG motifs.
I leak  g leak (Vi  Eleak )

I syn   g ( E
j 1
f (a, b,V ) 
(Vi ) 
 Vi )  g ( E
Ф2
V

1
1  exp[a(V  b)]
1
1  exp[1000(Vi  syn )
• The motif is strictly driven by inhibitory signals,
which are varied in strength.
• All neurons in the motif are configured identically
(see table of parameters).
• The burst regimes exhibit subtle distortions until gji ≈
0.66 (inset C), where a sub-region suddenly appears in the
green burst rhythm region, and continues to expand until
it becomes tangent to the line Ф1= Ф2 (insets D, E).
• At gji = 0.69 (inset F), another region appears, and this
process cascades at an increasing rate until gji=0.705 (inset
H), when regions fully desynchronized burst rhythms
appear (shown in gray) and the synchronized regions begin
to collapse (inset I).
E
F
H
I
Ф2
G
Ф2
• By gij=0.78 desynchronization occurs everywhere for all
phase points (Ф1, Ф2) (voltage trace, inset 4).
C = 0.5
GK2 = 30
EK = -0.07
ENa = 0.045
GNa = 160
GI = 8
EI = -0.046
Ipol = 0.006
σm = 0.0035
σh = 0.0065
τK2 = 0.9
τNa = 0.0405
Esyn = -0.0625
Θsyn = -0.03
n = 0.018
h = 0.99
Membrane capacitance, μF
K+ maximal conductance, nS/μm2
K+ reversal potential, V
Na+ reversal potential, V
Na+ maximal conductance, nS/μm2
leak maximal conductance, nS/μm2
leak reversal potential, V
polarization current, mA
2 Asymmetric inhibition:
• In the case of weakly inhibition, burst rhythms take significantly
longer to stabilize, allowing us to see the manner of convergence
to the final burst pattern outcome.
K time constant
Na time constant
inhibitory reversal potential, V
Synaptic threshold, V
Gating parameter for activation of IK
Gating parameter for inactivation of Ina
Cells send inhibitory signals of
equal strength in both directions.
3
• Each cell in the motifs has a pair of coupling strengths
gij ; i≠j.
• Different burst rhythms occur depending on the duration
of the phase shifts (Ф1 , Ф2). For instance at (Ф1 , Ф2) =
(0.6, 0.3) neuron 1 is anti-phase with respect to neurons 2
and 3.
• Due to inhibitory coupling, the motif gives rise to a
network period which may differ from the isolated
period of a single cell.
• With respect to neuron 1 (blue) we introduce a pair of
phase shifts (Ф1 , Ф2) which reflect the duration of time
that cells 2 and 3 are “off”, respectively. The shifts are
normalized with respect to the isolated period Tiso.
Ф1
Ф1
• We plot Φ1(t), Φ2(t), parameterized with respect to time t, indicating
the relative phase difference between neuron pairs (blue, green) and
(blue, red) respectively. This plot suggests possible mechanics of how
the bursting rhythm arrives at the synchronization state.
Further Research
• We intend to investigate the dynamics that give rise to the cascading burst rhythms for the strongly coupled
cases. Additionally, anti-phase (but not necessarily aperiodic) states should yield a series of attractors as well,
although these have not been characterized in the work shown.
• The basins of attraction for the weakly coupled case are significantly different from the strongly coupled case.
One possible way to observe this change in dynamics would be to identify the coupling strengths g ij where the
system tends from a weakly coupled motif to a strongly coupled one.
• The evolution of the dynamics of asymmetric weakly coupled motif are not yet known but may yield insight into
the bifurcations that give rise to the dynamics described thus far.
Ф2
Ф2
Ф1
Symmetric inhibition:
Cells send significantly stronger
inhibitory signals in one direction.
Ф1
Weak Coupling: Symmetric Motifs
3-Cell Inhibitory Networks
• The measurements made with regard to phase shift are isochronic, i.e. Φ1 and Φ2 are discretized with respect to
the isolated period. Because of this, more phase shift values are evaluated during the “slow” portion of burst
cycle (quiescence) than the “fast” portion (tonic spiking). To rectify this, it has been proposed that the isolated
periodic orbit be spliced into equal intervals, from which the phase shift time values would be interpolated.
Ф2
Ф1
Ф1
VK2shift  0.019; gij  0.0005
• Applying phase shifting to mixed CPG motifs (where inhibitory and excitatory signals are passed) would
potentially yield understanding of the complex dynamics generated by those networks.
• For weakly coupled motif (gij=0.0005), there exists well
defined regions of both in-phase and desynchronized
bursting states.
• The parameterized phase plot illustrates boundaries where
choices of Φ1 and Φ2 lead to a specific bursting rhythm,
which can be thought of as a stable fixed point in (Φ1, Φ2).
• Unstable and saddle activity occurs around the triangular
shaped gray regions corresponding to desynchronized burst
rhythms.
• The inset below depicts a voltage trace at (Φ1, Φ2) = (0.78,
0.31), where desynchronization occurs.
References
Ф2
Ф2
Ф1
Φ2 = 0.3
[1] Shilnikov, A. L., Rene, G., Belykh, I. (2008). Polyrhythmic synchronization in bursting networking
motifs. Chaos 18 pp 1-13.
Ф1
[2] Jalil, S., Belykh, I., Shilnikov, A. (2009). Synchronized bursting: the evil twin of the half-center
oscillator. PNAS, paper pending.
[3] Cymbalyuk, G. S., Calabrese, R. L., and Shilnikov, A. L. (2005). How a neuron model can demonstrate
co-existence of tonic spiking and bursting? Neurocomputing 65–66 , pp 869–875.
• Due to symmetric coupling, the regions
shown above are symmetric with respect to
the line Φ1= Φ2.
Tiso ≈11.31
V
Φ1 = 0.6
V
• Since the phase shifts are of unit modulus,
the phase shift plot can be thought of as
being on a torus (shown right), where
convergence to bursting rhythms (i.e. stable
fixed points) occurs along the surface.
V
Tcoup ≈ 12.02
t (ms)
D
 0.02; gij  - 0.1
• Also of notice is the appearance of unstable focus surrounded by three saddle nodes. Our current hypothesis of
the transitioning dynamics is that by shortening the burst (via increasing VK2shift ) the unstable focus will become
stable, by way of all three saddles collapsing onto the focus.
• Inhibitory coupling strengths are fixed in the clockwise
direction for gij ={g21, g32, g13}; gji={g12, g23, g31} are varied
identically in increasing magnitude from 0.1 to 0.9.
 Vi ) (V j   syn )
• For our purposes the excitatory coupling strengths
gexc=0 for all neurons in the motif.
1
• Very specific dynamics arise when the phase portrait for the symmetrically coupled case gij = 0.0005 is computed.
There exist 3 stable fixed points corresponding to the known burst rhythm outcomes where one cell is in antiphase with respect to the others. More notably, there exists a repeller at the origin, which suggests that unless the
phase shift is identically (0,0), the burst pattern will always tend to one of the regions (or otherwise be
desynchronized).
Ф1
Vi is the membrane potential,
INa is the sodium current,
IK is the potassium current,
Ileak is the leak current,
Ipol is the polarization current,
Isyn is the synaptic current,
gij is synaptic coupling strength between neurons i and j,
Γ is the sigmoid coupling function used to drive
inhibitory synaptic coupling between neurons
I K 2  g K 2 mi2 (Vi  EK )
exc
syn
C
where:
dV
C i  ( I Na  I K 2  I leak  I pol  I syn )
dt
dh
 Na i  f (500,0.026  h ,Vi )  hi
dt
dm
 K 2 i  f (83,0.018 VKshift
2 , Vi )  mi
dt
I Na  g Na f (150,0.27   n , Vi )3 hi (Vi  E Na )
exc
ij
B
• By varying the strength of the asymmetric coupling in the strongly coupled motif, we observe bursting regimes
that ultimately cascade into desynchronized burst rhythms.
• In order to observe the attractors and repellers of the phase system, we utilize a weak coupling motif that
produces a slower rate of synchronization between the burst patterns within the network.
3
shift
K2
The Hodgkin Huxley formulation for the (pharmacologically reduced) leech heart interneuron model is given as:
in
syn
4: (Ф1, Ф2)
= (0.8, 0.5);
gji = 0.8
Ф2
Leech Heart Interneuron Model
in
ij
2: (Ф1, Ф2)
= (0.4, 0.9)
A
2
• This research is focused on the onset of polyrhythmic dynamics in a model of a multifunctional CPG. Every
oscillatory attractor of the network corresponds to a specific rhythm and is conjectured to be associated with a
particular type of locomotive activity of a CPG. By elaborating on various configurations of mutually inhibitory and
mixed motifs, network building blocks, we intend to describe some universal synergetic mechanisms of emergent
synchronous behaviors in CPGs.
• Each burst rhythm that can be produced by the CPG functions as a oscillatory attractor of the system with respect
to the phase shifts of each cell.
• Convergence to each outcome is rapid, often after
the first burst cycle completes. +
1
n
3: (Ф1, Ф2)
= (0.8, 0.3)
1: (Ф1, Ф2)
= (0.2, 0.25)
Results and Discussion
t (ms)
t (ms)
[4] Nowotny, T., and Rabinovich, M. I. (2007). Dynamical Origin of Independent Spiking and Bursting
Activity in Neural Microcircuits. Phys. Rev. Letters 98, 128107.
[5] Ashwin, P., Burylko, O., Maistrenko, Y. (2008). Bifurcation to heteroclinic cycles and sensitivity in three
and four coupled phase oscillators. Physica D 237, pp 454-466.
[6] Cymbalyuk, G., Shilnikov, A. (2005). Coexistence of tonic spiking oscillations in a leech neuron model.
J. Comp. Neurosci. 18, pp 255-263.