Hello everyone!
I wanted to let you know about the recent publication of a Biophysical
Journal paper by Oprisan, S.A., V. Thirumalai, and C.C. Canavier, which
you might find interesting.
In addition, there are two other companion papers addressing similar
issues ("soft" perturbations of nonlinear oscillators and stability
analyses of couple ring network):
2) The influence of limit cycle topology on the phase resetting curve,
Oprisan S.A., and C.C. Canavier, Neural Computation 14(5):1027-1057, 2002.
3) Stability Analysis of Rings of Pulse-Coupled Oscillators: The Effect
of Phase Resetting in the Second Cycle After the Pulse Is Important at
Synchrony and For Long Pulses, Oprisan S.A., and C.C. Canavier, Journal
of Differential Equations and Dynamical Systems 9(3-4):243—258, 2001.
I'm attaching the abstracts here.
Please let me know if you would like reprints (electronic or hard copy)
and do not have access to these journals.
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Dynamics from a Time Series: Can We Extract the Phase Resetting Curve
from a Time Series?, Biophys. J. 2003 84: 2919-2928,
S. A. Oprisan*, V. Thirumalai**, and C. C. Canavier*
* Department of Psychology, University of New Orleans, New Orleans,
Louisiana
** Volen Center for Complex Systems, Brandeis University, Waltham,
Massachusetts
Recordings of the membrane potential from a bursting neuron were used to
reconstruct the phase curve for that neuron for a limited set of
perturbations. These perturbations were inhibitory synaptic conductance
pulses able to shift the membrane potential below the most
hyperpolarized level attained in the free running mode. The extraction
of the phase resetting curve from such a one-dimensional time series
requires reconstruction of the periodic activity in the form of a limit
cycle attractor. Resetting was found to have two components. In the
first component, if the pulse was applied during a burst, the burst was
truncated, and the time until the next burst was shortened in a manner
predicted by movement normal to the limit cycle. By movement normal to
the limit cycle, we mean a switch between two well-defined solution
branches of a relaxation-like oscillator in a hysteretic manner enabled
by the existence of a singular dominant slow process (variable). In the
second component, the onset of the burst was delayed until the end of
the hyperpolarizing pulse. Thus, for the pulse amplitudes we studied,
resetting was independent of amplitude but increased linearly with pulse
duration. The predicted and the experimental phase resetting curves for
a pyloric dilator neuron show satisfactory agreement. The method was
applied to only one pulse per cycle, but our results suggest it could
easily be generalized to accommodate multiple inputs.
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The Influence of Limit Cycle Topology on the Phase Resetting Curve
Sorinel A. Oprisan
soprisan@uno.edu, Department of Psychology, University of New Orleans,
New Orleans, LA 70148, U.S.A.
Carmen C. Canavier
ccanavie@uno.edu, Department of Psychology, University of New Orleans,
New Orleans, LA 70148, U.S.A.
Understanding the phenomenology of phase resetting is an essential step
toward developing a formalism for the analysis of circuits composed of
bursting neurons that receive multiple, and sometimes overlapping,
inputs. If we are to use phase-resetting methods to analyze these
circuits, we can either generate phase-resetting curves (PRCs) for all
possible inputs and combinations of inputs, or we can develop an
understanding of how to construct PRCs for arbitrary perturbations of a
given neuron. The latter strategy is the goal of this study.
We present a geometrical derivation of phase resetting of neural limit
cycle oscillators in response to short current pulses. A geometrical
phase is defined as the distance traveled along the limit cycle in the
appropriate phase space. The perturbations in current are treated as
displacements in the direction corresponding to membrane voltage. We
show that for type I oscillators, the direction of a perturbation in
current is nearly tangent to the limit cycle; hence, the projection of
the displacement in voltage onto the limit cycle is sufficient to give
the geometrical phase resetting. In order to obtain the phase resetting
in terms of elapsed time or temporal phase, a mapping between
geometrical and temporal phase is obtained empirically and used to make
the conversion. This mapping is shown to be an invariant of the
dynamics. Perturbations in current applied to type II oscillators
produce significant normal displacements from the limit cycle, so the
difference in angular velocity at displaced points compared to the
angular velocity on the limit cycle must be taken into account.
Empirical attempts to correct for differences in angular velocity
(amplitude versus phase effects in terms of a circular coordinate
system) during relaxation back to the limit cycle achieved some success
in the construction of phase-resetting curves for type II model
oscillators. The ultimate goal of this work is the extension of these
techniques to biological circuits comprising type II neural oscillators,
which appear frequently in identified central pattern-generating circuits.
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Stability Analysis of Rings of Pulse-Coupled Oscillators:
The Effect of Phase Resetting in the Second Cycle After the Pulse
Is Important at Synchrony and For Long Pulses
S.A. Oprisan and C.C. Canavier
Department of Psychology, University of New Orleans, New Orleans, LA 70148
Rings of limit cycle oscillators coupled in a pulsatile manner
to simulate the chemical synaptic coupling of neural oscillators
are approximated by discrete maps based on a phase resetting analysis.
Linearization about a fixed pattern of phase resetting is employed
to predict the stability of patterns such as synchrony or alternation.
Novel results are obtained with respect to synchrony in a two-neuron
ring, and previous results regarding nonsynchronous firing patterns
are extended to include the effects of phase resetting on
two consecutive cycles, rather than just one. The effects on two cycles
were considered because we wish to apply our results so circuits
in which the pulses last a significant fraction of the cycle, and can easily
affect two cycles if they occur late in the cycle.
**********
Best wishes,
Sorinel Oprisan
--
Dr. Sorinel Adrian OPRISAN
Department of Psychology, University of New Orleans
2000 Lakeshore Dr., New Orleans, LA 70148
Phone: (504)-280-6851, Fax: (504)-280-6049
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