Oscillatory reaction-diffusion equations on rings

We study the behavior of traveling waves in - systems on both homogeneous and inhomogeneous rings. The stability regions in parameter space of - waves were previously known [15, 19]; the results are extended here. We show the existence of Hopf bifurcations of traveling waves and the stability of the limit cycles born at the Hopf bifurcation for some parameter ranges. Using a Lindstedt-type perturbation scheme, we formally construct periodic solutions of the - system near a Hopf bifurcation and show that the periodic solutions superimposed on the original traveling wave have the effect of altering its overall frequency and amplitude. We also study the - system on an annulus ofvariable width, which does not possess reflection symmetry about any axis. We formally construct traveling waves on this variable-width annulus by a perturbation scheme, and find that perturbing the width of the annulus alters the amplitude and frequency of traveling waves on the domain by a small (order ²) amount. For typical parameter values, we find that the speed, frequency, and stability are unaffected by the direction of travel of the wave on the annulus, despite the rotationally asymmetric inhomogeneity. This indicates that the - system on a variable-width domain cannot account for directional preferences of traveling waves in biological systems.     

A Discrete Velocity Kinetic Model with Food Metric: Chemotaxis Traveling Waves

We introduce a mesoscopic scale chemotaxis model for traveling wave phenomena which is induced by food metric. The organisms of this simplified kinetic model have two discrete velocity modes, \(\pm s\) and a constant tumbling rate. The main feature of the model is that the speed of organisms is constant \(s\,{>}\,0\) with respect to the food metric, not the Euclidean metric. The uniqueness and the existence of the traveling wave solution of the model are obtained. Unlike the classical logarithmic model case there exist traveling waves under super-linear consumption rates and infinite population pulse-type traveling waves are obtained. Numerical simulations are also provided.     

A biologically motivated and analytically soluble model of collective oscillations in the cortex

A model of an associative network of spiking neurons with stationary states, globally locked oscillations, and weakly locked oscillatory states is presented and analyzed. The network is close to biology in the following sense. First, the neurons spike and our model includes an absolute refractory period after each spike. Second, we consider a distribution of axonal delay times. Finally, we describe synaptic signal transmission by excitatory and inhibitory potentials (EPSP and IPSP) with a realistic shape, that is, through a response kernel. During retrieval of a pattern, all active neurons exhibit periodic spike bursts which may or may not be synchronized (locked) into a coherent oscillation. We derive an analytical condition of locking and calculate the period of collective activity during oscillatory retrieval. In a stationary retrieval state, the overlap assumes a constant value proportional to the mean firing rate of the neurons. It is argued that in a biological network an intermediate scenario of weak locking is most likely.     

The nature of the coupling between segmental oscillators of the lamprey spinal generator for locomotion: A mathematical model

We present a theoretical model which is used to explain the intersegmental coordination of the neural networks responsible for generating locomotion in the isolated spinal cord of lamprey.A simplified mathematical model of a limit cycle oscillator is presented which consists of only a single dependent variable, the phase (t). By coupling N such oscillators together we are able to generate stable phase locked motions which correspond to traveling waves in the spinal cord, thus simulating fictive swimming. We are also able to generate irregular drifting motions which are compared to the experimental data obtained from cords with selective surgical lesions.     

Reasoning of spike glycoproteins being more vulnerable to mutations among 158 coronavirus proteins from different species

In this study, we used the probabilistic models developed by us over the last several years to analyze 158 proteins from coronaviruses in order to determine which protein is more vulnerable to mutations. The results provide three lines of evidence suggesting that the spike glycoprotein is different from the other coronavirus proteins: (1) the spike glycoprotein is more sensitive to mutations, this is the current state of the spike glycoprotein, (2) the spike glycoprotein has undergone more mutations in the past, this is the history of spike glycoprotein, and (3) the spike glycoprotein has a bigger potential towards future mutations, this is the future of spike glycoprotein. Furthermore, this study gives a clue on the species susceptibility regarding different proteins.Figure Predictable and unpredictable portions in coronavirus proteins. The data are presented as median with interquartile range. * the predictable and unpredictable portions in spike glycoprotein group are statistically different from any other protein groups at p<0.05 level, except for hemagglutinin-esterase precursor group. # the predictable and unpredictable portions in spike glycoprotein group are statistically different from hemagglutinin-esterase precursor, membrane protein and nucleocapsid protein groups at p<0.05 level. the predictable and unpredictable portions in spike glycoprotein group are statistically different from hemagglutinin-esterase precursor, and membrane protein groups at p<0.05 level.Electronic Supplementary Material is available for this article if you access the article at .     

Ionic mechanisms of electrical activity in somatic muscle of the nematodeAscaris lumbricoides

Summary The ionic dependence of the myogenic spike potentials and slow waves recorded fromAscaris lumbricoides somatic muscles has been investigated. Spikes appear to be mediated exclusively by calcium ions; the spike active potential varies with calcium concentration as expected for a calcium electrode and spikes persist in sodium-free media (Fig. 2). Slow waves can be mediated either by sodium or calcium; they persist when calcium or sodium are removed separately, but not when both are removed together (Figs. 3, 4, 6).In rhythmically active preparations, a burst of slow waves and spikes accompanies each contraction. Two phenomena may be related to the mechanism of this modulation:1) TEA, although it does not prolong slow waves or spikes, induces rhythmic bursts of activity similar to spontaneous modulation (Fig. 5). This TEA-induced modulation appears to be myogenic. 2) Under conditions where calcium influx is reduced (either by addition of EGTA to the bath or by replacement of calcium with barium or strontium), very long-duration square waves are observed (Figs. 4. 7. 8). The square waves resemble slow waves in their ionic dependence, but differ in their sensitivity to TEA and to variation in the external potassium concentration. It is suggested that modulation and square waves involve the same channels. The significance of these results in understanding the role of myogenic activity in nematode locomotion is discussed.We thank Mr. Mac McGlaughlin for help in obtainingAscaris. This work was supported by a Sloan Foundation grant in Neuroscience and a U.S. Public Health Service grant (NS 09654) to R.L.R., by an NIH Traineeship on grant BCH Tol GM 01262-12 to D.A.W., and by an NIH Postdoctoral Fellowship (1 FO2 GM55347) to L.B.     

Spike Timing-Dependent Synaptic Plasticity in Visual Cortex: A Modeling Study

Recent studies show that synaptic modification depends critically on the relative spike timing of pre- and postsynaptic neurons. Here we explore the functional implications of spike timing-dependent synaptic plasticity in the visual cortex using a model circuit with modifiable intracortical excitatory connections. First we simulated the experiments using two-point stimuli, in which two visual stimuli in a topographically represented feature space were repeatedly presented in quick succession, and found that tuning of the cortical neurons was modified in a manner similar to that observed experimentally. We then explored the dependence of results on the model parameter and identified the intracortical parameters that were critical for the magnitude of the shifts and obtained a simple relationship between the amount of shift and (S = (_EXTC^rec_exc)/_INHC^rec_inh). Finally we investigated the effects of moving stimuli in a topographically represented visual space and found that they can effectively induce spike timing-dependent modification of the intracortical connections. It suggests the importance of moving stimuli in dynamic modification of the cortical maps through spike timing-dependent synaptic plasticity.     

Cyclic competition of three species in the time periodic and diffusive case

We consider a semilinear reaction diffusion system with time periodic coefficients. The system models the competitive interaction of three species, which inhabit a bounded domain. We make assumptions that may be loosely stated (cyclically for i {1, 2, 3}) as: Species i outcompetes species i + 1 in the absence of species i + 2. Under those assumptions we prove the existence of a time periodic solution, which is strictly positive in each of its three components. Moreover, we obtain a new result on the nonexistence of positive time-periodic solutions and the extinction of one species for the related two-species subsystem. Our discussion includes a situation, known as cyclic competition in the autonomous ODE-case, in a more general framework that includes temporal and spatial heterogeneity.     

Dynamics of the Instantaneous Firing Rate in Response to Changes in Input Statistics

We review and extend recent results on the instantaneous firing rate dynamics of simplified models of spiking neurons in response to noisy current inputs. It has been shown recently that the response of the instantaneous firing rate to small amplitude oscillations in the mean inputs depends in the large frequency limit f on the spike initiation dynamics. A particular simplified model, the exponential integrate-and-fire (EIF) model, has a response that decays as 1/f in the large frequency limit and describes very well the response of conductance-based models with a Hodgkin-Huxley type fast sodium current. Here, we show that the response of the EIF instantaneous firing rate also decays as 1/f in the case of an oscillation in the variance of the inputs for both white and colored noise. We then compute the initial transient response of the firing rate of the EIF model to a step change in its mean inputs and/or in the variance of its inputs. We show that in both cases the response speed is proportional to the neuron stationary firing rate and inversely proportional to a spike slope factor _T that controls the sharpness of spike initiation: as 1/_T for a step change in mean inputs, and as 1/_T² for a step change in the variance in the inputs.     

Model predictions of the ionic mechanisms underlying the beating and bursting pacemaker characteristics of molluscan neurons

The general properties of the excitable membrane on molluscan pacemaker neurons can be described on the basis of a fair amount of experimental evidence available in the literature. The neuronal membrane exhibits under voltage clamp an initial inward current carried by both Na⁺ and Ca²⁺ ions, the time- and voltage-dependent characteristics of which are similar to that of other excitable structures. The conductance mechanism for the two ion species and the transport kinetics appear to be closely similar. The time course and amplitude of the delayed outward current carried by K⁺ ions shows a marked dependence on the membrane potential. Characteristic for the molluscan neurons is the existence of an additional fast transient outward current which is only activated by hyperpolarizing shifts from the membrane potential. A regular beating discharge over a wide range of frequencies can be predicted by making the assumption of a metabolically controlled driving of the Na⁺ conductance. Bursting pacemaker characteristics can be correctly simulated by the model if sinusoidal variations of an additional Na⁺ and Ca²⁺ conductances g _Na and g _Ca, and periodic variations of the K⁺ conductance g _K, governed by the known operation of a metabolic substrate cycle are introduced. The close approximation of experimentally observed impulse bursts requires that the actual inpulse-frequency and the amplitude of the after-spike hyperpolarization are determined by the temporal pattern of g _Na, while the spike amplitude is controlled by g _Na which (although of similar time course) is lagging in phase behing g _Na. The periodic changes in additional K⁺ conductance g _K, are responsible for burst termination and the changes in inter-burst interval, to the effect that spike doublets, triplets and multi-spike bursts can be simulated by a suitable choice for the time characteristics of g _K. The model makes use of the finding that the Ca²⁺ inflow associated with a spike discharge actually activates g _K, so that large postburst hyperpolarizations can be obtained in high-frequency bursts.Supported by the Deutsche Forschungsgemeinschaft (Grant Ch 25/1)     

Waves in a simple,excitable or oscillatory,reaction-diffusion model

A simple one variable caricature for oscillating and excitable reaction-diffusion systems is introduced. It is shown that as a parameter, , varies the system dynamics change from oscillatory ( > ₀) to excitable ( < ₀) and the frequency of the oscillation vanishes as for ₀. When such dynamics are coupled by continuous diffusion in a ring geometry (1-space dimension), propagating wave trains may be found. On an infinite ring excitable devices lead to unique solitary waves which are analogous to pulse waves. A solvable example is presented, illustrating properties of dispersion, excitability, and waves. Finally it is shown that the caricature arises in a natural way from more general excitable/oscillatory systems.     

Firing rhythmicity in the neocortex in vivo and in vitro under sustained iontophoretic excitation

Extracellular recordings were made from the cat intact neocortex and guinea-pig neocortical slices during microiontophoretic application of amino acid neurotransmitters. Spike train autocorrelation analysis showed a high stability of firing patterns in the intact neocortex. When excitation of a cell was increased in a step-wise manner with glutamate iontophoresis only an enhancement of the rate of firing was observed. The rhythmic component, which was mainly due to periodic multiple discharges, remained up to the highest firing frequencies. In contrast to the in vivo observation, glutamate, aspartate or K⁺ iontophoresis in cortical slices resulted in firing pattern alternations (always from bursts or irregular activity to regular spike firing) as well as an increase in firing rate. In slices the periodic component was typically due to single-spike regularity and its frequency rose with an increase of firing rate. The comparison of autocorrelogram alternations in vivo and in vitro suggests that the temporal organization of spike trains in the intact cortex is under tight external control and is defined mainly by neuronal interactions, whereas virtually all the neurons in vitro are very sensitive to the same iontophoretic influences and their individual outputs easily change according to the excitation (depolarization) level. The coincidence of the lowest frequencies of single-spike regularity in the in vitro preparation (5–7 Hz and 8–10 Hz) with theta- and alpha-rhythms in the electroencephalogram (EEG), and with single unit firing rhythmicity in the whole brain, may represent the basis of a unit-circuit resonance and provide a high stability of these EEG-rhythms.Abbreviations ACF autocorrelation function - BFA background firing activity - EEG electroencephalogram     

Existence and stability of periodic travelling wave solutions to Nagumo's nerve equation

Summary Nagumo's nerve conduction equation has a one-parameter family of spatially periodic travelling wave solutions. First, we prove the existence of these solutions by using a topological method. (There are some exceptional cases in which this method cannot be applied in showing the existence.) A periodic travelling wave solution corresponds to a closed orbit of a third-order dynamical system. The Poincaré index of the closed orbit is determined as a direct consequence of the proof of the existence. Second, we prove that the periodic travelling wave solution is unstable if the Poincaré index of the corresponding closed orbit is + 1. By using this result, together with the result of the author's previous paper, it is concluded that the slow periodic travelling wave solutions are always unstable. Third, we consider the stability of the fast periodic travelling wave solutions. We denote by L(c) the spatial period of the travelling wave solution with the propagation speed c. It is shown that the fast solution is unstable if its period is close to L_min, the minimum of L(c).     

Do calcium buffers always slow down the propagation of calcium waves?

Calcium buffers are large proteins that act as binding sites for free cytosolic calcium. Since a large fraction of cytosolic calcium is bound to calcium buffers, calcium waves are widely observed under the condition that free cytosolic calcium is heavily buffered. In addition, all physiological buffered excitable systems contain multiple buffers with different affinities. It is thus important to understand the properties of waves in excitable systems with the inclusion of buffers. There is an ongoing controversy about whether or not the addition of calcium buffers into the system always slows down the propagation of calcium waves. To solve this controversy, we incorporate the buffering effect into the generic excitable system, the FitzHugh–Nagumo model, to get the buffered FitzHugh–Nagumo model, and then to study the effect of the added buffer with large diffusivity on traveling waves of such a model in one spatial dimension. We can find a critical dissociation constant ( $K=K(a)$ ) characterized by system excitability parameter $a$ such that calcium buffers can be classified into two types: weak buffers ( $K\in (K(a),\infty )$ ) and strong buffers ( $K\in (0,K(a))$ ). We analytically show that the addition of weak buffers or strong buffers but with its total concentration $b_0^1$ below some critical total concentration $b_{0,c}^1$ into the system can generate a traveling wave of the resulting system which propagates faster than that of the origin system, provided that the diffusivity $D_1$ of the added buffers is sufficiently large. Further, the magnitude of the wave speed of traveling waves of the resulting system is proportional to $\sqrt{D_1}$ as $D_1\rightarrow \infty $ . In contrast, the addition of strong buffers with the total concentration $b_0^1>b_{0,c}^1$ into the system may not be able to support the formation of a biologically acceptable wave provided that the diffusivity $D_1$ of the added buffers is sufficiently large.     

Threshold quantities for helminth infections

For parasites with a clearly defined life-cycle we give threshold quantities that determine the stability of the parasite-free steady state for autonomous and periodic deterministic systems formulated in terms of mean parasite burdens. We discuss the biological interpretations of the quantities, how to deal with heterogeneity in both parasite and host populations, how to incorporate the effects of periodic discontinuities, and the relation of the threshold quantities to the basic reproduction ratio R ₀. Examples from the literature are given. The analysis of the periodic case extends easily to micro-parasitic systems.     

Pooled spike trains of correlated presynaptic inputs as realizations of cluster point processes

The pooled spike trains of correlated presynaptic terminals acting synchronously upon a single neuron are realizations of cluster point processes: the notions of spikes synchronizing in bursts and of points bunching in clusters are conceptually identical. The primary processes constituent specifies the timing of the cluster series; subsidiary processes and poolings specify burst structure and tightness. This representation and the Poisson process representation of independent terminals complete the formal approach to pooled trains. The notions usefulness was illustrated by expressing physiological questions in terms of those constituents, each possessing a clear biological embodiment; constituents provided the control variables in simulations using leaky integrate-and-fire postsynaptic neurons excited by multiple weak terminals. Regular or irregular primary processes and bursts series determined low or high postsynaptic dispersions. When convergent set synchrony increased, its postsynaptic consequences approached those of single powerful synapses; concomitantly, output spike trains approached periodic, quasiperiodic, or aperiodic behaviors. The sequence in which terminals fired within bursts affected the predictee and predictor roles of presynaptic and postsynaptic spikes; when inhibition was added, EPSP and IPSP delays and order were influential (summation was noncommutative). Outputs to different correlations were heterogeneous; heterogeneity was accentuated by conditioning by variables such as DC biases.     

Subthreshold outward currents enhance temporal integration in auditory neurons

Many auditory neurons possess low-threshold potassium currents (I _KLT ) that enhance their responsiveness to rapid and coincident inputs. We present recordings from gerbil medial superior olivary (MSO) neurons in vitro and modeling results that illustrate how I _KLT improves the detection of brief signals, of weak signals in noise, and of the coincidence of signals (as needed for sound localization). We quantify the enhancing effect of I _KLT on temporal processing with several measures: signal-to-noise ratio (SNR), reverse correlation or spike-triggered averaging of input currents, and interaural time difference (ITD) tuning curves. To characterize how I _KLT , which activates below spike threshold, influences a neurons voltage rise toward threshold, i.e., how it filters the inputs, we focus first on the response to weak and noisy signals. Cells and models were stimulated with a computer-generated steady barrage of random inputs, mimicking weak synaptic conductance transients (the noise), together with a larger but still subthreshold postsynaptic conductance, EPSG (the signal). Reduction of I _KLT decreased the SNR, mainly due to an increase in spontaneous firing (more false positive). The spike-triggered reverse correlation indicated that I _KLT shortened the integration time for spike generation. I _KLT also heightened the models timing selectivity for coincidence detection of simulated binaural inputs. Further, ITD tuning is shifted in favor of a slope code rather than a place code by precise and rapid inhibition onto MSO cells (Brand et al. 2002). In several ways, low-threshold outward currents are seen to shape integration of weak and strong signals in auditory neurons.     

Sound production through the substrate during reproduction in the mottled sculpin,Cottus bairdi (Cottidae)

Synopsis During reproduction maleCottus bairdi defend cavities beneath stones and perform defense and reproductive displays. Using a geophone to detect substrate vibrations under dark conditions (infrared viewing), we recorded three types of sounds. Knocks are produced during head nods and an acoustically similar sound is produced when the fish slaps the head to the substrate. A third sound, the drum roll appears to be a fast repetition of several knocks followed by a head slap. We argue that these signals traveling through the substrate are of greater importance than sounds traveling through the water because (1) the substrate vibration attenuates at a much lesser rate than the water vibration and, (2) even near riffles, which generate much water vibration, the background noise in the substrate is low enough for the fish to detect such sounds.     

Observations on phase-locking within the response of primary muscle spindle afferents to pseudo-random stretch

In order to uncover encoder properties of primary muscle spindle afferent fibers, time coupling (phase-locking) of action potentials on cyclic muscle stretch was studied by means of pseudo-random noise. In cats Ia action potentials were recorded from dorsal root filaments and the gastrocnemius muscles of one hind leg were stretched. The stimulus time course was a determined sequence of randomly varying muscle length which could be applied repeatedly (sequence duration 0.6 or 20 s). The noise amplitude (standard deviation of displacements) was varied between 5 and 300 m, the upper cut-off frequency of noise f _c was varied between 20 and 100 Hz. The responses to the consecutive pseudo-random noise cycles were displayed as raster diagrams and cycle histograms. Phaselocking characterized the responses at all noise amplitudes outside the near threshold range (>10 m). The higher and f _c , the stronger was the phase-locking of impulses on the stretch. When and f _c were selected to achieve high mean stretch velocities of about 500 mm/s, phase-locking was as precise as 0.15 ms, measured as the variability of spike occurrences with respect to stretch. The rasters obtained with low noise amplitudes (<40 m) showed a loose phase-locking and this gave insight into underlying mechanisms: The elicitation of action potentials caused by dynamic stretch can be prevented by a post-spike depression of excitability. This disfacilitation was very effective in counteracting weak stretch components within the random sequence and less effective or even missing when relatively strong stretch components could force the spike elicitation. This led to the reestablishment of phase-locked patterns. The results were discussed in relation to the known encoder models.