72 research outputs found
Spike-Timing Dependent Plasticity and Feed-Forward Input Oscillations Produce Precise and Invariant Spike Phase-Locking
In the hippocampus and the neocortex, the coupling between local field potential (LFP) oscillations and the spiking of single neurons can be highly precise, across neuronal populations and cell types. Spike phase (i.e., the spike time with respect to a reference oscillation) is known to carry reliable information, both with phase-locking behavior and with more complex phase relationships, such as phase precession. How this precision is achieved by neuronal populations, whose membrane properties and total input may be quite heterogeneous, is nevertheless unknown. In this note, we investigate a simple mechanism for learning precise LFP-to-spike coupling in feed-forward networks – the reliable, periodic modulation of presynaptic firing rates during oscillations, coupled with spike-timing dependent plasticity. When oscillations are within the biological range (2–150 Hz), firing rates of the inputs change on a timescale highly relevant to spike-timing dependent plasticity (STDP). Through analytic and computational methods, we find points of stable phase-locking for a neuron with plastic input synapses. These points correspond to precise phase-locking behavior in the feed-forward network. The location of these points depends on the oscillation frequency of the inputs, the STDP time constants, and the balance of potentiation and de-potentiation in the STDP rule. For a given input oscillation, the balance of potentiation and de-potentiation in the STDP rule is the critical parameter that determines the phase at which an output neuron will learn to spike. These findings are robust to changes in intrinsic post-synaptic properties. Finally, we discuss implications of this mechanism for stable learning of spike-timing in the hippocampus
On a link between Dirichlet kernels and central multinomial coefficients
AbstractThe central coefficients of powers of certain polynomials with arbitrary degree in x form an important family of integer sequences. Although various recursive equations addressing these coefficients do exist, no explicit analytic representation has yet been proposed. In this article, we present an explicit form of the integer sequences of central multinomial coefficients of polynomials of even degree in terms of finite sums over Dirichlet kernels, hence linking these sequences to discrete nth-degree Fourier series expansions. The approach utilizes the diagonalization of circulant Boolean matrices, and is generalizable to all multinomial coefficients of certain polynomials with even degree, thus forming the base for a new family of combinatorial identities
Theory of transient chimeras in finite Sakaguchi-Kuramoto networks
Chimera states are a phenomenon in which order and disorder can co-exist
within a network that is fully homogeneous. Precisely how transient chimeras
emerge in finite networks of Kuramoto oscillators with phase-lag remains
unclear. Utilizing an operator-based framework to study nonlinear oscillator
networks at finite scale, we reveal the spatiotemporal impact of the adjacency
matrix eigenvectors on the Sakaguchi-Kuramoto dynamics. We identify a specific
condition for the emergence of transient chimeras in these finite networks: the
eigenvectors of the network adjacency matrix create a combination of a zero
phase-offset mode and low spatial frequency waves traveling in opposite
directions. This combination of eigenvectors leads directly to the coherent and
incoherent clusters in the chimera. This approach provides two specific
analytical predictions: (1) a precise formula predicting the combination of
connectivity and phase-lag that creates transient chimeras, (2) a mathematical
procedure for rewiring arbitrary networks to produce transient chimeras
Composed solutions of synchronized patterns in multiplex networks of Kuramoto oscillators
Networks with different levels of interactions, including multilayer and
multiplex networks, can display a rich diversity of dynamical behaviors and can
be used to model and study a wide range of systems. Despite numerous efforts to
investigate these networks, obtaining mathematical descriptions for the
dynamics of multilayer and multiplex systems is still an open problem. Here, we
combine ideas and concepts from linear algebra and graph theory with nonlinear
dynamics to offer a novel approach to study multiplex networks of Kuramoto
oscillators. Our approach allows us to study the dynamics of a large, multiplex
network by decomposing it into two smaller systems: one representing the
connection scheme within layers (intra-layer), and the other representing the
connections between layers (inter-layer). Particularly, we use this approach to
compose solutions for multiplex networks of Kuramoto oscillators. These
solutions are given by a combination of solutions for the smaller systems given
by the intra and inter-layer system and, in addition, our approach allows us to
study the linear stability of these solutions
Small changes at single nodes can shift global network dynamics
Understanding the sensitivity of a system's behavior with respect to
parameter changes is essential for many applications. This sensitivity may be
desired - for instance in the brain, where a large repertoire of different
dynamics, particularly different synchronization patterns, is crucial - or may
be undesired - for instance in power grids, where disruptions to
synchronization may lead to blackouts. In this work, we show that the dynamics
of networks of phase oscillators can acquire a very large and complex
sensitivity to changes made in either their units' parameters or in their
connections - even modifications made to a parameter of a single unit can
radically alter the global dynamics of the network in an unpredictable manner.
As a consequence, each modification leads to a different path to phase
synchronization manifested as large fluctuations along that path. This
dynamical malleability occurs over a wide parameter region, around the
network's two transitions to phase synchronization. One transition is induced
by increasing the coupling strength between the units, and another is induced
by increasing the prevalence of long-range connections. Specifically, we study
Kuramoto phase oscillators connected under either Watts-Strogatz or
distance-dependent topologies to analyze the statistical properties of the
fluctuations along the paths to phase synchrony. We argue that this increase in
the dynamical malleability is a general phenomenon, as suggested by both
previous studies and the theory of phase transitions.Comment: 14 pages, 8 figure
State-dependent activity dynamics of hypothalamic stress effector neurons
The stress response necessitates an immediate boost in vital physiological functions from their homeostatic operation to an elevated emergency response. However, the neural mechanisms underlying this state-dependent change remain largely unknown. Using a combination of in vivo and ex vivo electrophysiology with computational modeling, we report that corticotropin releasing hormone (CRH) neurons in the paraventricular nucleus of the hypothalamus (PVN), the effector neurons of hormonal stress response, rapidly transition between distinct activity states through recurrent inhibition. Specifically, in vivo optrode recording shows that under non-stress conditions, CRHPVN neurons often fire with rhythmic brief bursts (RB), which, somewhat counterintuitively, constrains firing rate due to long (~2 s) interburst intervals. Stressful stimuli rapidly switch RB to continuous single spiking (SS), permitting a large increase in firing rate. A spiking network model shows that recurrent inhibition can control this activity-state switch, and more broadly the gain of spiking responses to excitatory inputs. In biological CRHPVN neurons ex vivo, the injection of whole-cell currents derived from our computational model recreates the in vivo-like switch between RB and SS, providing direct evidence that physiologically relevant network inputs enable state-dependent computation in single neurons. Together, we present a novel mechanism for state-dependent activity dynamics in CRHPVN neurons
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