M-type potassium conductance controls the emergence of neural phase codes: a combined experimental and neuron modelling study

Rate and phase codes are believed to be important in neural information processing. Hippocampal place cells provide a good example where both coding schemes coexist during spatial information processing. Spike rate increases in the place field, whereas spike phase precesses relative to the ongoing theta oscillation. However, what intrinsic mechanism allows for a single neuron to generate spike output patterns that contain both neural codes is unknown. Using dynamic clamp, we simulate an in vivo-like subthreshold dynamics of place cells to in vitro CA1 pyramidal neurons to establish an in vitro model of spike phase precession. Using this in vitro model, we show that membrane potential oscillation (MPO) dynamics is important in the emergence of spike phase codes: blocking the slowly activating, non-inactivating K⁺ current (I_M), which is known to control subthreshold MPO, disrupts MPO and abolishes spike phase precession. We verify the importance of adaptive I_M in the generation of phase codes using both an adaptive integrate-and-fire and a Hodgkin–Huxley (HH) neuron model. Especially, using the HH model, we further show that it is the perisomatically located I_M with slow activation kinetics that is crucial for the generation of phase codes. These results suggest an important functional role of I_M in single neuron computation, where I_M serves as an intrinsic mechanism allowing for dual rate and phase coding in single neurons.     

Unsupervised learning of control signals and their encodings in Caenorhabditis elegans whole-brain recordings

A major goal of computational neuroscience is to understand the relationship between synapse-level structure and network-level functionality. Caenorhabditis elegans is a model organism to probe this relationship due to the historic availability of the synaptic structure (connectome) and recent advances in whole brain calcium imaging techniques. Recent work has applied the concept of network controllability to neuronal networks, discovering some neurons that are able to drive the network to a certain state. However, previous work uses a linear model of the network dynamics, and it is unclear if the real neuronal network conforms to this assumption. Here, we propose a method to build a global, low-dimensional model of the dynamics, whereby an underlying global linear dynamical system is actuated by temporally sparse control signals. A key novelty of this method is discovering candidate control signals that the network uses to control itself. We analyse these control signals in two ways, showing they are interpretable and biologically plausible. First, these control signals are associated with transitions between behaviours, which were previously annotated via expert-generated features. Second, these signals can be predicted both from neurons previously implicated in behavioural transitions but also additional neurons previously unassociated with these behaviours. The proposed mathematical framework is generic and can be generalized to other neurosensory systems, potentially revealing transitions and their encodings in a completely unsupervised way.     

Circadian regulation of sleep-wake behaviour in nocturnal rats requires multiple signals from suprachiasmatic nucleus

The dynamics of sleep and wake are strongly linked to the circadian clock. Many models have accurately predicted behaviour resulting from dynamic interactions between these two systems without specifying physiological substrates for these interactions. By contrast, recent experimental work has identified much of the relevant physiology for circadian and sleep-wake regulation, but interaction dynamics are difficult to study experimentally. To bridge these approaches, we developed a neuronal population model for the dynamic, bidirectional, neurotransmitter-mediated interactions of the sleep-wake and circadian regulatory systems in nocturnal rats. This model proposes that the central circadian pacemaker, located within the suprachiasmatic nucleus (SCN) of the hypothalamus, promotes sleep through single neurotransmitter-mediated signalling to sleep-wake regulatory populations. Feedback projections from these populations to the SCN alter SCN firing patterns and fine-tune this modulation. Although this model reproduced circadian variation in sleep-wake dynamics in nocturnal rats, it failed to describe the sleep-wake dynamics observed in SCN-lesioned rats. We thus propose two alternative, physiologically based models in which neurotransmitter- and neuropeptide-mediated signalling from the SCN to sleep-wake populations introduces mechanisms to account for the behaviour of both the intact and SCN-lesioned rat. These models generate testable predictions and offer a new framework for modelling sleep-wake and circadian interactions.     

Towards blueprints for network architecture, biophysical dynamics and signal transduction

We review mathematical aspects of biophysical dynamics, signal transduction and network architecture that have been used to uncover functionally significant relations between the dynamics of single neurons and the networks they compose. We focus on examples that combine insights from these three areas to expand our understanding of systems neuroscience. These range from single neuron coding to models of decision making and electrosensory discrimination by networks and populations and also coincidence detection in pairs of dendrites and dynamics of large networks of excitable dendritic spines. We conclude by describing some of the challenges that lie ahead as the applied mathematics community seeks to provide the tools which will ultimately underpin systems neuroscience.     

Dynamics of a hybrid thermostat model with discrete sampling time control

The dynamics of a simple thermostat model is described. In the model the control system samples the temperature at regular but discrete time intervals rather than by continuous monitoring. The model exhibits quasi-periodic oscillations and banding, where the response falls into two or more bands of phase space representing either better or poorer control. A return circle map is derived which explains the observed dynamics. Some extensions of these results to the case where the flow is non-linear are also given.     

Glial progenitor cell recruitment drives aggressive glioma growth: mathematical and experimental modelling

Currently available glioma treatments remain unsuccessful at prolonging disease-free remission. Recent evidence suggests that tumour recruitment of glial progenitor cells by platelet-derived growth factor (PDGF) may play a role in the development and progression of these tumours. Building upon our recent experimental results and previous proliferation–invasion (PI) reaction–diffusion model, in this study, we created a proliferation–invasion–recruitment (PIR) model that includes a mechanism for progenitor cell recruitment, wherein paracrine PDGF signalling stimulates migration and proliferation of progenitors derived from the local brain environment. Parametrizing this mathematical model with data obtained from the PDGF-driven rat glioma model, we explored the consequences of recruitment, using the PIR model to compare the effects of high versus low PDGF secretion rates on tumour growth and invasion dynamics. The mathematical model predicts correlation between high levels of recruitment and both increased radial velocity of expansion on magnetic resonance imaging and less diffusely invasive edges. Thus, the PIR model predicts that PDGF levels correlate with tumour aggressiveness, and results are consistent with both human and experimental data, demonstrating that the effects of progenitor cell recruitment provide a novel mechanism to explain the variability in the rates of proliferation and dispersion observed in human gliomas.     

Excitation and suppression of oscillations in a chain of unidirectionally coupled elements with neuron-like dynamics

The propagation of pulses in a model chain of Hindmarsh-Rose neurons with unidirectional coupling has been studied in the presence of intrinsic dynamics in chain elements. The main regimes of oscillations in the model chain are determined as dependent on the coupling parameter. The mechanisms of activation and suppression of oscillations in the model chain under the action of rectangular pulses are explained using the results of a detailed analysis of the basins of attraction of a stable immobile point and a periodic attractor of a model neuron.     

Chaotic oscillations of two cascade-coupled oscillators with frequency control

The generation of chaotically modulated oscillations in autooscillatory systems with frequency control is considered. It is shown that, by coupling two such systems, it is possible to considerably expand a region corresponding to the generation of chaotic oscillations in the space of parameters of this ensemble and to provide additional means of controlling the characteristics of oscillations by changing the parameters of coupling.     

Collaboration structures in Slovenian scientific communities

We combine two seemingly distinct perspectives regarding the modeling of network dynamics. One perspective is found in the work of physicists and mathematicians who formally introduced the small world model and the mechanism of preferential attachment. The other perspective is sociological and focuses on the process of cumulative advantage and considers the agency of individual actors in a network. We test hypotheses, based on work drawn from these perspectives, regarding the structure and dynamics of scientific collaboration networks. The data we use are for four scientific disciplines in the Slovene system of science. The results deal with the overall topology of these networks and specific processes that generate them. The two perspectives can be joined to mutual benefit. Within this combined approach, the presence of small-world structures was confirmed. However preferential attachment is far more complex than advocates of a single autonomous mechanism claim.     

Generation of periodic high-power ultrashort pulse sequences in a chain of coupled traveling-wave tubes operating in the regimes of amplification and nonlinear Kompfner suppression

We have studied the dynamics of an electron microwave oscillator with a feedback loop containing two coupled traveling-wave tubes, the first operating in the regime of amplification and the second in the regime of nonlinear Kompfner absorption. It is established that oscillations generated in this system can take the form of a periodic sequence of ultrashort pulses. The proposed mechanism of pulse generation is analogous to the well-known method of passive mode locking.     

Hybrid discrete-time neural networks

Hybrid dynamical systems combine evolution equations with state transitions. When the evolution equations are discrete-time (also called map-based), the result is a hybrid discrete-time system. A class of biological neural network models that has recently received some attention falls within this category: map-based neuron models connected by means of fast threshold modulation (FTM). FTM is a connection scheme that aims to mimic the switching dynamics of a neuron subject to synaptic inputs. The dynamic equations of the neuron adopt different forms according to the state (either firing or not firing) and type (excitatory or inhibitory) of their presynaptic neighbours. Therefore, the mathematical model of one such network is a combination of discrete-time evolution equations with transitions between states, constituting a hybrid discrete-time (map-based) neural network. In this paper, we review previous work within the context of these models, exemplifying useful techniques to analyse them. Typical map-based neuron models are low-dimensional and amenable to phase-plane analysis. In bursting models, fast-slow decomposition can be used to reduce dimensionality further, so that the dynamics of a pair of connected neurons can be easily understood. We also discuss a model that includes electrical synapses in addition to chemical synapses with FTM. Furthermore, we describe how master stability functions can predict the stability of synchronized states in these networks. The main results are extended to larger map-based neural networks.     

Features of the spatiotemporal activity switching in a two-dimensional array of Rose-Hindmarsh model neurons

Features of the spatiotemporal dynamics in a two-dimensional array of Rose-Hindmarsh model neurons with inhomogeneous coupling have been studied. A function is proposed for evaluating the rate of spiking and bursting activity in such systems with two characteristic scales of oscillations. Based on this function, the influence of the inhomogeneity of coupling on the system dynamics is analyzed.     

Contribution of time delays to p53 oscillation in DNA damage response

Although the oscillatory dynamics of the p53 network have been extensively studied, the understanding of the mechanism of delay‐induced oscillations is still limited. In this paper, a comprehensive mathematical model of p53 network is studied, which contains two delayed negative feedback loops. By studying the model with and without explicit delays, the results indicate that the time delay of Mdm2 protein synthesis can well control the pulse shape but cannot induce p53 oscillation alone, while the time delay required for Wip1 protein synthesis induces a Hopf bifurcation to drive p53 oscillation. In addition, the synergy of the two delays will cause the p53 network to oscillate in advance, indicating that p53 begins the repair process earlier in the damaged cell. Furthermore, the stability and bifurcation of the model are addressed, which may highlight the role of time delay in p53 oscillations.Inspec keywords: proteins, cellular biophysics, DNA, molecular biophysics, biomolecular effects of radiation, bifurcation, physiological models, cellular effects of radiation, oscillations, geneticsOther keywords: highlight, time delay, delayed negative feedback loops, murine double minute 2, Wip1 protein synthesis, explicit delays, Mdm2 protein synthesis, p53 network     

Converting harmonic oscillations into chaos

A new simple model of a system with chaotic dynamics, based on the equations of bistable systems, is considered. The possibility of converting harmonic signals into chaotic oscillations, which represent intermittent irregular and switching quasi-regular motions, is demonstrated by numerical methods. The mechanism of chaotization is analyzed using the results of numerical calculations.     

嗅球中神经元间的相互作用及同步运动分析

嗅球对嗅觉信息的处理是嗅觉系统信号编码的一个重要环节,其中兴奋性的僧帽细胞（Mitral Cell, MC）与抑制性的颗粒细胞（Granular Cell, GC）的相互作用尤为关键。本文首先介绍了嗅觉系统网络中关于同步振荡的研究现状,然后建立嗅球中僧帽细胞及颗粒细胞的动力学模型,仿真得到了单个僧帽细胞、颗粒细胞以及僧帽细胞与颗粒细胞在耦合条件下神经元的发放模式。结果表明,僧帽细胞对颗粒细胞有兴奋性作用,而颗粒细胞对僧帽细胞有抑制性作用,细胞放电序列随着突触连接强度的改变而改变。此外,建立简单的嗅觉网络模型,分析了当颗粒细胞分别构成环形和网格状两种拓扑结构时,不同网络对两个僧帽细胞同步性的影响,用同步性指标ISI-distance刻画同步程度。数值分析表明颗粒细胞网格状的拓扑结构对僧帽细胞的同步性作用更为明显一些。     

Dramatic Effect of Vibrations on Dynamics of Rotating Hydrodynamic Systems

The results of theoretical and experimental investigation of the average dynamics of inhomogeneous systems in rotating cavity in the presence of oscillating force field are highlighted and analyzed. Average behavior of nonisothermal liquid as well as different multiphase systems (two liquids, liquid and gas, solid in liquid, granular matter in liquid) in quickly rotating horizontal cylinder subjected to vibrations of different direction is described. The specificity of dynamics of all the systems is determined by the Coriolis force which defines not only the average flows but the fluid oscillations also. The resonant excitation of inertial oscillations of liquid results in generation of intensive mean flows. The structure of vibrational streams is essentially determined by the type of inertial waves. So, excitation of an azimuthal wave generates azimuthal streams (outrunning or lagging, depending on frequency of the wave), three-dimensional standing waves (in the systems with deformable interface) generate regular spatial flows of high intensity. Under the resonant conditions the influence of vibrations appears to be very strong, the speed of mean streaming can achieve the values comparable with the speed of rotation of the container.     

Hyperbolic chaos generator based on coupled drift klystrons

A scheme of the generator of chaotic oscillations in the microwave band is proposed that is based on two drift klystrons coupled in a ring circuit. The first klystron doubles the frequency of the input signal and the second klystron mixes the second harmonic signal and the reference signal representing a periodic sequence of radio pulses filled with the third harmonic, followed by separation of the differential first harmonic signal. As a result, the transformation of the signal phase during the period of pulse repetition is described by a stretching map of the circle (Bernoulli map) and demonstrates chaotic behavior. Results of numerical calculations are presented that confirm the implementation of the proposed mechanism to ensure the generation of robust, structurally stable chaos in the system under consideration.     

Non-homogeneous extracellular resistivity affects the current-source density profiles of up-down state oscillations

Rhythmic local field potential (LFP) oscillations observed during deep sleep are the result of synchronized electrical activities of large neuronal ensembles, which consist of alternating periods of activity and silence, termed 'up' and 'down' states, respectively. Current-source density (CSD) analysis indicates that the up states of these slow oscillations are associated with current sources in superficial cortical layers and sinks in deep layers, while the down states display the opposite pattern of source-sink distribution. We show here that a network model of up and down states displays this CSD profile only if a frequency-filtering extracellular medium is assumed. When frequency filtering was modelled as inhomogeneous conductivity, this simple model had considerably more power in slow frequencies, resulting in significant differences in LFP and CSD profiles compared with the constant-resistivity model. These results suggest that the frequency-filtering properties of extracellular media may have important consequences for the interpretation of the results of CSD analysis.     

Mesoscopic systems: classical irreversibility and quantum coherence

Mesoscopic physics is a sub-discipline of condensed-matter physics that focuses on the properties of solids in a size range intermediate between bulk matter and individual atoms. In particular, it is characteristic of a domain where a certain number of interacting objects can easily be tuned between classical and quantum regimes, thus enabling studies at the border of the two. In magnetism, such a tuning was first realized with large-spin magnetic molecules called single-molecule magnets (SMMs) with archetype Mn(12)-ac. In general, the mesoscopic scale can be relatively large (e.g. micrometre-sized superconducting circuits), but, in magnetism, it is much smaller and can reach the atomic scale with rare earth (RE) ions. In all cases, it is shown how quantum relaxation can drastically reduce classical irreversibility. Taking the example of mesoscopic spin systems, the origin of irreversibility is discussed on the basis of the Landau-Zener model. A classical counterpart of this model is described enabling, in particular, intuitive understanding of most aspects of quantum spin dynamics. The spin dynamics of mesoscopic spin systems (SMM or RE systems) becomes coherent if they are well isolated. The study of the damping of their Rabi oscillations gives access to most relevant decoherence mechanisms by different environmental baths, including the electromagnetic bath of microwave excitation. This type of decoherence, clearly seen with spin systems, is easily recovered in quantum simulations. It is also observed with other types of qubits such as a single spin in a quantum dot or a superconducting loop, despite the presence of other competitive decoherence mechanisms. As in the molecular magnet V(15), the leading decoherence terms of superconducting qubits seem to be associated with a non-Markovian channel in which short-living entanglements with distributions of two-level systems (nuclear spins, impurity spins and/or charges) leading to 1/f noise induce τ(1)-like relaxation of S(z) with dissipation to the bath of two-level systems with which they interact most. Finally, let us mention that these experiments on quantum oscillations are, most of the time, performed in the classical regime of Rabi oscillations, suggesting that decoherence mechanisms might also be treated classically.