814 research outputs found
Wild oscillations in a nonlinear neuron model with resets: (II) Mixed-mode oscillations
This work continues the analysis of complex dynamics in a class of
bidimensional nonlinear hybrid dynamical systems with resets modeling neuronal
voltage dynamics with adaptation and spike emission. We show that these models
can generically display a form of mixed-mode oscillations (MMOs), which are
trajectories featuring an alternation of small oscillations with spikes or
bursts (multiple consecutive spikes). The mechanism by which these are
generated relies fundamentally on the hybrid structure of the flow: invariant
manifolds of the continuous dynamics govern small oscillations, while discrete
resets govern the emission of spikes or bursts, contrasting with classical MMO
mechanisms in ordinary differential equations involving more than three
dimensions and generally relying on a timescale separation. The decomposition
of mechanisms reveals the geometrical origin of MMOs, allowing a relatively
simple classification of points on the reset manifold associated to specific
numbers of small oscillations. We show that the MMO pattern can be described
through the study of orbits of a discrete adaptation map, which is singular as
it features discrete discontinuities with unbounded left- and
right-derivatives. We study orbits of the map via rotation theory for
discontinuous circle maps and elucidate in detail complex behaviors arising in
the case where MMOs display at most one small oscillation between each
consecutive pair of spikes
On Dynamics of Integrate-and-Fire Neural Networks with Conductance Based Synapses
We present a mathematical analysis of a networks with Integrate-and-Fire
neurons and adaptive conductances. Taking into account the realistic fact that
the spike time is only known within some \textit{finite} precision, we propose
a model where spikes are effective at times multiple of a characteristic time
scale , where can be \textit{arbitrary} small (in particular,
well beyond the numerical precision). We make a complete mathematical
characterization of the model-dynamics and obtain the following results. The
asymptotic dynamics is composed by finitely many stable periodic orbits, whose
number and period can be arbitrary large and can diverge in a region of the
synaptic weights space, traditionally called the "edge of chaos", a notion
mathematically well defined in the present paper. Furthermore, except at the
edge of chaos, there is a one-to-one correspondence between the membrane
potential trajectories and the raster plot. This shows that the neural code is
entirely "in the spikes" in this case. As a key tool, we introduce an order
parameter, easy to compute numerically, and closely related to a natural notion
of entropy, providing a relevant characterization of the computational
capabilities of the network. This allows us to compare the computational
capabilities of leaky and Integrate-and-Fire models and conductance based
models. The present study considers networks with constant input, and without
time-dependent plasticity, but the framework has been designed for both
extensions.Comment: 36 pages, 9 figure
Entrainment and chaos in a pulse-driven Hodgkin-Huxley oscillator
The Hodgkin-Huxley model describes action potential generation in certain
types of neurons and is a standard model for conductance-based, excitable
cells. Following the early work of Winfree and Best, this paper explores the
response of a spontaneously spiking Hodgkin-Huxley neuron model to a periodic
pulsatile drive. The response as a function of drive period and amplitude is
systematically characterized. A wide range of qualitatively distinct responses
are found, including entrainment to the input pulse train and persistent chaos.
These observations are consistent with a theory of kicked oscillators developed
by Qiudong Wang and Lai-Sang Young. In addition to general features predicted
by Wang-Young theory, it is found that most combinations of drive period and
amplitude lead to entrainment instead of chaos. This preference for entrainment
over chaos is explained by the structure of the Hodgkin-Huxley phase resetting
curve.Comment: Minor revisions; modified Fig. 3; added reference
Neuron as a reward-modulated combinatorial switch and a model of learning behavior
This paper proposes a neuronal circuitry layout and synaptic plasticity
principles that allow the (pyramidal) neuron to act as a "combinatorial
switch". Namely, the neuron learns to be more prone to generate spikes given
those combinations of firing input neurons for which a previous spiking of the
neuron had been followed by a positive global reward signal. The reward signal
may be mediated by certain modulatory hormones or neurotransmitters, e.g., the
dopamine. More generally, a trial-and-error learning paradigm is suggested in
which a global reward signal triggers long-term enhancement or weakening of a
neuron's spiking response to the preceding neuronal input firing pattern. Thus,
rewards provide a feedback pathway that informs neurons whether their spiking
was beneficial or detrimental for a particular input combination. The neuron's
ability to discern specific combinations of firing input neurons is achieved
through a random or predetermined spatial distribution of input synapses on
dendrites that creates synaptic clusters that represent various permutations of
input neurons. The corresponding dendritic segments, or the enclosed individual
spines, are capable of being particularly excited, due to local sigmoidal
thresholding involving voltage-gated channel conductances, if the segment's
excitatory and absence of inhibitory inputs are temporally coincident. Such
nonlinear excitation corresponds to a particular firing combination of input
neurons, and it is posited that the excitation strength encodes the
combinatorial memory and is regulated by long-term plasticity mechanisms. It is
also suggested that the spine calcium influx that may result from the
spatiotemporal synaptic input coincidence may cause the spine head actin
filaments to undergo mechanical (muscle-like) contraction, with the ensuing
cytoskeletal deformation transmitted to the axon initial segment where it
may...Comment: Version 5: added computer code in the ancillary files sectio
Optimal Population Codes for Space: Grid Cells Outperform Place Cells
Rodents use two distinct neuronal coordinate systems to estimate their position: place fields in the hippocampus and grid fields in the entorhinal cortex. Whereas place cells spike at only one particular spatial location, grid cells fire at multiple sites that correspond to the points of an imaginary hexagonal lattice. We study how to best construct place and grid codes, taking the probabilistic nature of neural spiking into account. Which spatial encoding properties of individual neurons confer the highest resolution when decoding the animal’s position from the neuronal population response? A priori, estimating a spatial position from a grid code could be ambiguous, as regular periodic lattices possess translational symmetry. The solution to this problem requires lattices for grid cells with different spacings; the spatial resolution crucially depends on choosing the right ratios of these spacings across the population. We compute the expected error in estimating the position in both the asymptotic limit, using Fisher information, and for low spike counts, using maximum likelihood estimation. Achieving high spatial resolution and covering a large range of space in a grid code leads to a trade-off: the best grid code for spatial resolution is built of nested modules with different spatial periods, one inside the other, whereas maximizing the spatial range requires distinct spatial periods that are pairwisely incommensurate. Optimizing the spatial resolution predicts two grid cell properties that have been experimentally observed. First, short lattice spacings should outnumber long lattice spacings. Second, the grid code should be self-similar across different lattice spacings, so that the grid field always covers a fixed fraction of the lattice period. If these conditions are satisfied and the spatial “tuning curves” for each neuron span the same range of firing rates, then the resolution of the grid code easily exceeds that of the best possible place code with the same number of neurons
On the interaction of gamma-rhythmic neuronal populations
Local gamma-band (~30-100Hz) oscillations in the brain, produced by feedback inhibition on a characteristic timescale, appear in multiple areas of the brain and are associated with a wide range of cognitive functions. Some regions producing gamma also receive gamma-rhythmic input, and the interaction and coordination of these rhythms has been hypothesized to serve various functional roles. This thesis consists of three stand-alone chapters, each of which considers the response of a gamma-rhythmic neuronal circuit to input in an analytical framework. In the first, we demonstrate that several related models of a gamma-generating circuit under periodic forcing are asymptotically drawn onto an attracting invariant torus due to the convergence of inhibition trajectories at spikes and the convergence of voltage trajectories during sustained inhibition, and therefore display a restricted range of dynamics. In the second, we show that a model of a gamma-generating circuit under forcing by square pulses cannot maintain multiple stably phase-locked solutions. In the third, we show that a separation of time scales of membrane potential dynamics and synaptic decay causes the gamma model to phase align its spiking such that periodic forcing pulses arrive under minimal inhibition. When two of these models are mutually coupled, the same effect causes excitatory pulses from the faster oscillator to arrive at the slower under minimal inhibition, while pulses from the slower to the faster arrive under maximal inhibition. We also show that such a time scale separation allows the model to respond sensitively to input pulse coherence to an extent that is not possible for a simple one-dimensional oscillator. We draw on a wide range of mathematical tools and structures including return maps, saltation matrices, contraction methods, phase response formalism, and singular perturbation theory in order to show that the neuronal mechanism of gamma oscillations is uniquely suited to reliably phase lock across brain regions and facilitate the selective transmission of information
The cerebellum could solve the motor error problem through error increase prediction
We present a cerebellar architecture with two main characteristics. The first
one is that complex spikes respond to increases in sensory errors. The second
one is that cerebellar modules associate particular contexts where errors have
increased in the past with corrective commands that stop the increase in error.
We analyze our architecture formally and computationally for the case of
reaching in a 3D environment. In the case of motor control, we show that there
are synergies of this architecture with the Equilibrium-Point hypothesis,
leading to novel ways to solve the motor error problem. In particular, the
presence of desired equilibrium lengths for muscles provides a way to know when
the error is increasing, and which corrections to apply. In the context of
Threshold Control Theory and Perceptual Control Theory we show how to extend
our model so it implements anticipative corrections in cascade control systems
that span from muscle contractions to cognitive operations.Comment: 34 pages (without bibliography), 13 figure
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