Theta oscillons in behaving rats.

Recently discovered constituents of the brain waves-the oscillons-provide high-resolution representation of the extracellular field dynamics. Here we study the most robust, highest-amplitude oscillons that manifest in actively behaving rats and generally correspond to the traditional θ-waves. We show that the resemblances between θ-oscillons and the conventional θ-waves apply to the ballpark characteristics-mean frequencies, amplitudes, and bandwidths. In addition, both hippocampal and cortical oscillons exhibit a number of intricate, behavior-attuned, transient properties that suggest a new vantage point for understanding the θ-rhythms' structure, origins and functions. We demonstrate that oscillons are frequency-modulated waves, with speed-controlled parameters, embedded into a noise background. We also use a basic model of neuronal synchronization to contextualize and to interpret the observed phenomena. In particular, we argue that the synchronicity level in physiological networks is fairly weak and modulated by the animal's locomotion.

Theta oscillons in behaving rats.

Recently discovered constituents of the brain waves-the oscillons-provide high-resolution representation of the extracellular field dynamics. Here we study the most robust, highest-amplitude oscillons that manifest in actively behaving rats and generally correspond to the traditional θ-waves. We show that the resemblances between θ-oscillons and the conventional θ-waves apply to the ballpark characteristics-mean frequencies, amplitudes, and bandwidths. In addition, both hippocampal and cortical oscillons exhibit a number of intricate, behavior-attuned, transient properties that suggest a new vantage point for understanding the θ-rhythms' structure, origins and functions. We demonstrate that oscillons are frequency-modulated waves, with speed-controlled parameters, embedded into a noise background. We also use a basic model of neuronal synchronization to contextualize and to interpret the observed phenomena. In particular, we argue that the synchronicity level in physiological networks is fairly weak and modulated by the animal's locomotion.

Altered patterning of neural activity in a tauopathy mouse model.

Alzheimer's disease (AD) is a complex neurodegenerative condition that manifests at multiple levels and involves a spectrum of abnormalities ranging from the cellular to cognitive. Here, we investigate the impact of AD-related tau-pathology on hippocampal circuits in mice engaged in spatial navigation, and study changes of neuronal firing and dynamics of extracellular fields. While most studies are based on analyzing instantaneous or time-averaged characteristics of neuronal activity, we focus on intermediate timescales-spike trains and waveforms of oscillatory potentials, which we consider as single entities. We find that, in healthy mice, spike arrangements and wave patterns (series of crests or troughs) are coupled to the animal's location, speed, and acceleration. In contrast, in tau-mice, neural activity is structurally disarrayed: brainwave cadence is detached from locomotion, spatial selectivity is lost, the spike flow is scrambled. Importantly, these alterations start early and accumulate with age, which exposes progressive disinvolvement the hippocampus circuit in spatial navigation. These features highlight qualitatively different neurodynamics than the ones provided by conventional analyses, and are more salient, thus revealing a new level of the hippocampal circuit disruptions.

Spatial representability of neuronal activity.

A common approach to interpreting spiking activity is based on identifying the firing fields-regions in physical or configuration spaces that elicit responses of neurons. Common examples include hippocampal place cells that fire at preferred locations in the navigated environment, head direction cells that fire at preferred orientations of the animal's head, view cells that respond to preferred spots in the visual field, etc. In all these cases, firing fields were discovered empirically, by trial and error. We argue that the existence and a number of properties of the firing fields can be established theoretically, through topological analyses of the neuronal spiking activity. In particular, we use Leray criterion powered by persistent homology theory, Eckhoff conditions and Region Connection Calculus to verify consistency of neuronal responses with a single coherent representation of space.

Discrete Structure of the Brain Rhythms.

Neuronal activity in the brain generates synchronous oscillations of the Local Field Potential (LFP). The traditional analyses of the LFPs are based on decomposing the signal into simpler components, such as sinusoidal harmonics. However, a common drawback of such methods is that the decomposition primitives are usually presumed from the onset, which may bias our understanding of the signal's structure. Here, we introduce an alternative approach that allows an impartial, high resolution, hands-off decomposition of the brain waves into a small number of discrete, frequency-modulated oscillatory processes, which we call oscillons. In particular, we demonstrate that mouse hippocampal LFP contain a single oscillon that occupies the θ-frequency band and a couple of γ-oscillons that correspond, respectively, to slow and fast γ-waves. Since the oscillons were identified empirically, they may represent the actual, physical structure of synchronous oscillations in neuronal ensembles, whereas Fourier-defined "brain waves" are nothing but poorly resolved oscillons.

Maintaining Consistency of Spatial Information in the Hippocampal Network: A Combinatorial Geometry Model.

Place cells in the rat hippocampus play a key role in creating the animal's internal representation of the world. During active navigation, these cells spike only in discrete locations, together encoding a map of the environment. Electrophysiological recordings have shown that the animal can revisit this map mentally during both sleep and awake states, reactivating the place cells that fired during its exploration in the same sequence in which they were originally activated. Although consistency of place cell activity during active navigation is arguably enforced by sensory and proprioceptive inputs, it remains unclear how a consistent representation of space can be maintained during spontaneous replay. We propose a model that can account for this phenomenon and suggest that a spatially consistent replay requires a number of constraints on the hippocampal network that affect its synaptic architecture and the statistics of synaptic connection strengths.

A topological paradigm for hippocampal spatial map formation using persistent homology.

An animal's ability to navigate through space rests on its ability to create a mental map of its environment. The hippocampus is the brain region centrally responsible for such maps, and it has been assumed to encode geometric information (distances, angles). Given, however, that hippocampal output consists of patterns of spiking across many neurons, and downstream regions must be able to translate those patterns into accurate information about an animal's spatial environment, we hypothesized that 1) the temporal pattern of neuronal firing, particularly co-firing, is key to decoding spatial information, and 2) since co-firing implies spatial overlap of place fields, a map encoded by co-firing will be based on connectivity and adjacency, i.e., it will be a topological map. Here we test this topological hypothesis with a simple model of hippocampal activity, varying three parameters (firing rate, place field size, and number of neurons) in computer simulations of rat trajectories in three topologically and geometrically distinct test environments. Using a computational algorithm based on recently developed tools from Persistent Homology theory in the field of algebraic topology, we find that the patterns of neuronal co-firing can, in fact, convey topological information about the environment in a biologically realistic length of time. Furthermore, our simulations reveal a "learning region" that highlights the interplay between the parameters in combining to produce hippocampal states that are more or less adept at map formation. For example, within the learning region a lower number of neurons firing can be compensated by adjustments in firing rate or place field size, but beyond a certain point map formation begins to fail. We propose that this learning region provides a coherent theoretical lens through which to view conditions that impair spatial learning by altering place cell firing rates or spatial specificity.

Exact, convergent periodic-orbit expansions of individual energy eigenvalues of regular quantum graphs.

We present exact, explicit, convergent periodic-orbit expansions for individual energy levels of regular quantum graphs in the paper. One simple application is the energy levels of a particle in a piecewise constant potential. Since the classical ray trajectories (including ray splitting) in such systems are strongly chaotic, this result provides an explicit quantization of a classically chaotic system.

Exact trace formulas for a class of one-dimensional ray-splitting systems.

Using quantum graph theory we establish that the ray-splitting trace formula proposed by Couchman et al. [Phys. Rev. A 46, 6193 (1992)] is exact for a class of one-dimensional ray-splitting systems. Important applications in combinatorics are suggested.

Multiple Perron-Frobenius operators.

A cycle expansion technique for discrete sums of several PF operators, similar to the one used in the standard classical dynamical zeta-function formalism is constructed. It is shown that the corresponding expansion coefficients show an interesting universal behavior, which illustrates the details of the interference between the particular mappings entering the sum.

Dynamical ansatz for path integrals and nonperturbative trace formulas.

It is shown that a recently discovered representation of the Green's function is equivalent to a certain "dynamical ansatz" for the corresponding path integral, which brings about a convenient method of nonperturbative approximations. Based on this observation, a set of nonperturbative approximations to the trace of the Green's function is established.