Phase Bursting Rhythms in Inhibitory Rings

1

Phase Bursting Rhythms in Inhibitory Rings
Matthew Brooks, Robert Clewley, and Andrey
Shilnikov







Results and Discussion
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. 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.
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. 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. 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 anti-phase 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). 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.
Burst rhythm outcomes are computed for
discretized values of phase pairs (?1, ?2) with
?1, ?2 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. Convergence to each outcome is
rapid, often after the first burst cycle
completes.
3 (?1, ?2) (0.8, 0.3)
1 (?1, ?2) (0.2, 0.25)
2 (?1, ?2) (0.4, 0.9)
4 (?1, ?2) (0.8, 0.5) gji 0.8
A
B
C
Leech Heart Interneuron Model
D
E
F
Inhibitory coupling strengths are fixed in the
clockwise direction for gij g21, g32, g13
gjig12, g23, g31 are varied identically in
increasing magnitude from 0.1 to 0.9. 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 ?1 ?2 (insets D, E).
At gji 0.69 (inset F), another region appears,
and this process cascades at an increasing rate
until gji0.705 (inset H), when regions fully
desynchronized burst rhythms appear (shown in
gray) and the synchronized regions begin to
collapse (inset I). By gij0.78
desynchronization occurs everywhere for all
phase points (?1, ?2) (voltage trace, inset 4).
The Hodgkin Huxley formulation for the
(pharmacologically reduced) leech heart
interneuron model is given as
where 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, G is the sigmoid
coupling function used to drive inhibitory
synaptic coupling between neurons
H
I
C 0.5 GK2 30 EK -0.07 ENa 0.045 GNa 160 GI 8 EI -0.046 Ipol 0.006 sm 0.0035 sh 0.0065 tK2 0.9 tNa 0.0405 Esyn -0.0625 Tsyn -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     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
Further Research

Weak Coupling Symmetric Motifs
For our purposes the excitatory coupling
strengths gexc0 for all neurons in the motif.
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).
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.
We plot F1(t), F2(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.
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 gij 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. The measurements made with
regard to phase shift are isochronic, i.e. F1 and
F2 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. 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.
3-Cell Inhibitory Networks
Asymmetric inhibition Cells send significantly
stronger inhibitory signals in one direction.
Symmetric inhibition Cells send inhibitory
signals of equal strength in both directions.
For weakly coupled motif (gij0.0005), there
exists well defined regions of both in-phase and
desynchronized bursting states. The
parameterized phase plot illustrates boundaries
where choices of F1 and F2 lead to a specific
bursting rhythm, which can be thought of as a
stable fixed point in (F1, F2). Unstable and
saddle activity occurs around the triangular
shaped gray regions corresponding to
desynchronized burst rhythms. The inset
below depicts a voltage trace at (F1, F2)
(0.78, 0.31), where desynchronization occurs.
References
Each cell in the motifs has a pair of coupling
strengths gij i?j. 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 (?1 ,
?2) 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.
Different burst rhythms occur depending on the
duration of the phase shifts (?1 , ?2). For
instance at (?1 , ?2) (0.6, 0.3) neuron 1 is
anti-phase with respect to neurons 2 and 3.
1 Shilnikov, A. L., Rene, G., Belykh, I.
(2008). Polyrhythmic synchronization in bursting
networking motifs. Chaos 18 pp 1-13. 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 6566 , pp
869875. 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.
Due to symmetric coupling, the regions shown
above are symmetric with respect to the line F1
F2. 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
t (ms)