Synchronization dynamics of two coupled neural oscillators receiving shared and unshared noisy stimuli

The response of neurons to external stimuli greatly depends on the intrinsic dynamics of the network. Here, the intrinsic dynamics are modeled as coupling and the external input is modeled as shared and unshared noise. We assume the neurons are repetitively firing action potentials (i.e., neural oscillators), are weakly and identically coupled, and the external noise is weak. Shared noise can induce bistability between the synchronous and anti-phase states even though the anti-phase state is the only stable state in the absence of noise. We study the Fokker-Planck equation of the system and perform an asymptotic reduction ρ ₀. The ρ ₀ solution is more computationally efficient than both the Monte Carlo simulations and the 2D Fokker-Planck solver, and agrees remarkably well with the full system with weak noise and weak coupling. With moderate noise and coupling, ρ ₀ is still qualitatively correct despite the small noise and coupling assumption in the asymptotic reduction. Our phase model accurately predicts the behavior of a realistic synaptically coupled Morris-Lecar system.

Theta phase coding in a network model of the entorhinal cortex layer II with entorhinal-hippocampal loop connections

We investigated successive firing of the stellate cells within a theta cycle, which replicates the phase coding of place information, using a network model of the entorhinal cortex layer II with loop connections. Layer II of the entorhinal cortex (ECII) sends signals to the hippocampus, and the hippocampus sends signals back to layer V of the entorhinal cortex (ECV). In addition to this major pathway, projection from ECV to ECII also exists. It is, therefore, inferred that reverberation activity readily appears if projections from ECV to ECII are potentiated. The frequency of the reverberation would be in a gamma range because it takes signals 20–30 ms to go around the entorhinal-hippocampal loop circuits. On the other hand, it has been suggested that ECII is a theta rhythm generator. If the reverberation activity appears in the entorhinal-hippocampal loop circuits, gamma oscillation would be superimposed on a theta rhythm in ECII like a gamma-theta oscillation. This is a reminiscence of the theta phase coding of place information. In this paper, first, a network model of ECII will be developed in order to reproduce a theta rhythm. Secondly, we will show that loop connections from one stellate cell to the other one are selectively potentiated by afferent signals to ECII. Frequencies of those afferent signals are different, and transmission delay of the loop connections is 20 ms. As a result, stellate cells fire successively within one cycle of the theta rhythm. This resembles gamma-theta oscillation underlying the phase coding. Our model also replicates the phase precession of stellate cell firing within a cycle of subthreshold oscillation (theta rhythm).     

Neural dynamics of the cognitive map in the hippocampus

The rodent hippocampus has been thought to represent the spatial environment as a cognitive map. In the classical theory, the cognitive map has been explained as a consequence of the fact that different spatial regions are assigned to different cell populations in the framework of rate coding. Recently, the relation between place cell firing and local field oscillation theta in terms of theta phase precession was experimentally discovered and suggested as a temporal coding mechanism leading to memory formation of behavioral sequences accompanied with asymmetric Hebbian plasticity. The cognitive map theory is apparently outside of the sequence memory view. Therefore, theoretical analysis is necessary to consider the biological neural dynamics for the sequence encoding of the memory of behavioral sequences, providing the cognitive map formation. In this article, we summarize the theoretical neural dynamics of the real-time sequence encoding by theta phase precession, called theta phase coding, and review a series of theoretical models with the theta phase coding that we previously reported. With respect to memory encoding functions, instantaneous memory formation of one-time experience was first demonstrated, and then the ability of integration of memories of behavioral sequences into a network of the cognitive map was shown. In terms of memory retrieval functions, theta phase coding enables the hippocampus to represent the spatial location in the current behavioral context even with ambiguous sensory input when multiple sequences were coded. Finally, for utilization, retrieved temporal sequences in the hippocampus can be available for action selection, through the process of reverting theta rhythm-dependent activities to information in the behavioral time scale. This theoretical approach allows us to investigate how the behavioral sequences are encoded, updated, retrieved and used in the hippocampus, as the real-time interaction with the external environment. It may indeed be the bridge to the episodic memory function in human hippocampus.     

Properties of synaptic noise in tonically active human motoneurons

The objective of these experiments was to determine the amount of synaptic noise on the cell membrane at various intervals after an action potential in a motoneuron firing at a specified frequency. Sources of noise such as variations in the level of voluntary drive were minimized by selecting only segments of the spike train in which the unit was running within prescribed frequency limits. The level of the membrane potential of the motoneuron during these intervals was determined using two test “pulses” (compound Ia excitatory postsynaptic potentials) of known amplitude. This enabled the probability of the membrane potential falling within a voltage “window” of known size at known times after the preceding spike to be determined. The probability density histograms showed that the fluctuations of membrane potential about a target interspike trajectory (i.e., the membrane noise) increased with time after the preceding spike. These fluctuations in the membrane potential can be accounted for by a one-dimensional “random walk” model of membrane noise. This model explains the salient features of the interval histograms, such as positive skewness at low target frequencies. A quantitative test of the model demonstrated its applicability to the motor pools of tibialis and masseter.     

Analysis of clustered firing patterns in synaptically coupled networks of oscillators

Oscillators in networks may display a variety of activity patterns. This paper presents a geometric singular perturbation analysis of clustering, or alternate firing of synchronized subgroups, among synaptically coupled oscillators. We consider oscillators in two types of networks: mutually coupled, with all-to-all inhibitory connections, and globally inhibitory, with one excitatory and one inhibitory population of oscillators, each of arbitrary size. Our analysis yields existence and stability conditions for clustered states, along with formulas for the periods of such firing patterns. By using two different approaches, we derive complementary conditions, the first set stated in terms of time lengths determined by intrinsic and synaptic properties of the oscillators and their coupling and the second set stated in terms of model parameters and phase space structures directly linked to parameters. These results suggest how biological components may interact to produce the spindle sleep rhythm in thalamocortical networks. Received: 9 September 1999 / Revised version: 7 July 2000 / Published online: 24 November 2000     

Multiple oscillations in changing cell shape by the plasmodium of Physarum polycephalum: general formula governing oscillatory phenomena by the Physarum plasmodium

The long-term dynamics of an amoeboid cell shape were studied using Physarum polycephalum plasmodia with various sizes. Cell shape varied oscillatorily in a multiple periodic manner. The organism periodically elongated with period of T₇ = 10 h, branched with T₆ = 4 h, became uneven with T₅ = 30 min and T₄ = 10 min, and blew up with T₃ = 1.5 min. Tiny plasmodia changed shape much faster with T₃ = 1.3 min, T₂ = 24 s and T₁ = 3.3 s simultaneously. The plasmodial cytoskeleton also showed periodic pattern formation with T₆, T₅ and T₃. Periods of all known oscillatory phenomena in this organism correspond to some of the periods for the above seven rhythms, and the following geometric progression holds among the periods: T_i + 1/T_i = 7 and T_i + 2/T_i + 1 = 3, where i = 1, 3, 5. Thus, multiple oscillations in the plasmodium are organized globally.     

Preparation of Parasagittal Slices for the Investigation of Dorsal-ventral Organization of the Rodent Medial Entorhinal Cortex

Computation in the brain relies on neurons responding appropriately to their synaptic inputs. Neurons differ in their complement and distribution of membrane ion channels that determine how they respond to synaptic inputs. However, the relationship between these cellular properties and neuronal function in behaving animals is not well understood. One approach to this problem is to investigate topographically organized neural circuits in which the position of individual neurons maps onto information they encode or computations they carry out¹. Experiments using this approach suggest principles for tuning of synaptic responses underlying information encoding in sensory and cognitive circuits^2,3.The topographical organization of spatial representations along the dorsal-ventral axis of the medial entorhinal cortex (MEC) provides an opportunity to establish relationships between cellular mechanisms and computations important for spatial cognition. Neurons in layer II of the rodent MEC encode location using grid-like firing fields^4-6. For neurons found at dorsal positions in the MEC the distance between the individual firing fields that form a grid is on the order of 30 cm, whereas for neurons at progressively more ventral positions this distance increases to greater than 1 m. Several studies have revealed cellular properties of neurons in layer II of the MEC that, like the spacing between grid firing fields, also differ according to their dorsal-ventral position, suggesting that these cellular properties are important for spatial computation^2,7-10.Here we describe procedures for preparation and electrophysiological recording from brain slices that maintain the dorsal-ventral extent of the MEC enabling investigation of the topographical organization of biophysical and anatomical properties of MEC neurons. The dorsal-ventral position of identified neurons relative to anatomical landmarks is difficult to establish accurately with protocols that use horizontal slices of MEC^7,8,11,12, as it is difficult to establish reference points for the exact dorsal-ventral location of the slice. The procedures we describe enable accurate and consistent measurement of location of recorded cells along the dorsal-ventral axis of the MEC as well as visualization of molecular gradients^2,10. The procedures have been developed for use with adult mice (> 28 days) and have been successfully employed with mice up to 1.5 years old. With adjustments they could be used with younger mice or other rodent species. A standardized system of preparation and measurement will aid systematic investigation of the cellular and microcircuit properties of this area.     

Synchronization properties of networks of electrically coupled neurons in the presence of noise and heterogeneities

We investigate how synchrony can be generated or induced in networks of electrically coupled integrate-and-fire neurons subject to noisy and heterogeneous inputs. Using analytical tools, we find that in a network under constant external inputs, synchrony can appear via a Hopf bifurcation from the asynchronous state to an oscillatory state. In a homogeneous net work, in the oscillatory state all neurons fire in synchrony, while in a heterogeneous network synchrony is looser, many neurons skipping cycles of the oscillation. If the transmission of action potentials via the electrical synapses is effectively excitatory, the Hopf bifurcation is supercritical, while effectively inhibitory transmission due to pronounced hyperpolarization leads to a subcritical bifurcation. In the latter case, the network exhibits bistability between an asynchronous state and an oscillatory state where all the neurons fire in synchrony. Finally we show that for time-varying external inputs, electrical coupling enhances the synchronization in an asynchronous network via a resonance at the firing-rate frequency.

Circadian changes in synaptic ribbons and spherules in pinealocytes of the Syrian hamster (Mesocricetus auratus)

Summary In the present study, synaptic ribbons were studied morphologically and quantitatively in hamster pineal gland. The number of ribbons and spherules of hamster pinealocytes was counted over a 24-h period. The 24-h variations in the quantity of synaptic ribbons were found to parallel fluctuations in pineal melatonin concentrations. No significant circadian changes were observed for synaptic spherules, indicating different roles for these two structures.     

Robust synchronization of coupled circadian and cell cycle oscillators in single mammalian cells

Circadian cycles and cell cycles are two fundamental periodic processes with a period in the range of 1 day. Consequently, coupling between such cycles can lead to synchronization. Here, we estimated the mutual interactions between the two oscillators by time‐lapse imaging of single mammalian NIH3T3 fibroblasts during several days. The analysis of thousands of circadian cycles in dividing cells clearly indicated that both oscillators tick in a 1:1 mode‐locked state, with cell divisions occurring tightly 5 h before the peak in circadian Rev‐Erbα‐YFP reporter expression. In principle, such synchrony may be caused by either unidirectional or bidirectional coupling. While gating of cell division by the circadian cycle has been most studied, our data combined with stochastic modeling unambiguously show that the reverse coupling is predominant in NIH3T3 cells. Moreover, temperature, genetic, and pharmacological perturbations showed that the two interacting cellular oscillators adopt a synchronized state that is highly robust over a wide range of parameters. These findings have implications for circadian function in proliferative tissues, including epidermis, immune cells, and cancer.     

LTD windows of the STDP learning rule and synaptic connections having a large transmission delay enable robust sequence learning amid background noise

Spike-timing-dependent synaptic plasticity (STDP) is a simple and effective learning rule for sequence learning. However, synapses being subject to STDP rules are readily influenced in noisy circumstances because synaptic conductances are modified by pre- and postsynaptic spikes elicited within a few tens of milliseconds, regardless of whether those spikes convey information or not. Noisy firing existing everywhere in the brain may induce irrelevant enhancement of synaptic connections through STDP rules and would result in uncertain memory encoding and obscure memory patterns. We will here show that the LTD windows of the STDP rules enable robust sequence learning amid background noise in cooperation with a large signal transmission delay between neurons and a theta rhythm, using a network model of the entorhinal cortex layer II with entorhinal-hippocampal loop connections. The important element of the present model for robust sequence learning amid background noise is the symmetric STDP rule having LTD windows on both sides of the LTP window, in addition to the loop connections having a large signal transmission delay and the theta rhythm pacing activities of stellate cells. Above all, the LTD window in the range of positive spike-timing is important to prevent influences of noise with the progress of sequence learning.     

Oscillating signals in the sensory pathway of halobacteria induced by periodic light stimuli

The swimming behaviour of Halobacterium salinarium can be modulated by light. Changes of the light intensity that induce reversals of the swimming direction are called repellent stimuli, those that suppress reversals, which otherwise would occur spontaneously from time to time, are called attractant stimuli. Bacteria were stimulated by periodic pulse-like stimuli, and the frequency of induced reversals was recorded. Stimulation with a period length between 16 and 6.5 s let the cells reverse periodically with the frequency of the external force. After the stimulation had been stopped, the cells continued to reverse periodically for 3 to 9 periods which, however, switched to a value of about 6 to 8 s, independent of the frequency of preceding stimulation. This endogeneous oscillation was most distinct when the stimulation period either equalled the endogeneous period or was twice or half of its length. During the endogeneous oscillation, the responsiveness to an attractant stimulus showed a pronounced phase-dependence. These results point to the oscillation of a signal in the sensory pathway which, different from our former assumption, seems to be not self-sustained but has to be set going by external stimulation. Received: 14 January 1998 / Revised version: 9 April 1998 / Accepted: 17 May 1998     

Numerical exploration of the influence of neural noise on the psychometric function at low stimulation intensity levels

The relationship between stimulus intensity and the probability of detecting the presence of the stimulus is described by the psychometrical function. The probabilistic nature of this relationship is based on the stochastic behaviour of sensory neural channels and sensory networks involved in perceptual processing (Kiang 1968). This study tries to establish a continuum of variability across different levels of integration in the central nervous system. Once the opening and closing times of ionic channels was simulated, a threshold to the collective behaviour of voltage-gated ionic channels was imposed in order to generate the spike train of a single neuron. Afterwards, the trains of spikes of different neurons were added up, simulating the activity of a sensory nerve. By adding the activity due to the stimulus to the spontaneous neural behaviour, the psychometric function was simulated using a thresholding approach. The results can replicate the stochastic resonance phenomenon, but also open up the possibility that attentional phenomena can be mediated not only by increasing neural activity (bursting or oscillatory), but also by increasing noise at the neural level.     

The dynamic structure underlying subthreshold oscillatory activity and the onset of spikes in a model of medial entorhinal cortex stellate cells

Medial entorhinal cortex layer II stellate cells display subthreshold oscillations (STOs). We study a single compartment biophysical model of such cells which qualitatively reproduces these STOs. We argue that in the subthreshold interval (STI) the seven-dimensional model can be reduced to a three-dimensional system of equations with well differentiated times scales. Using dynamical systems arguments we provide a mechanism for generations of STOs. This mechanism is based on the “canard structure,” in which relevant trajectories stay close to repelling manifolds for a significant interval of time. We also show that the transition from subthreshold oscillatory activity to spiking (“canard explosion”) is controlled in the STI by the same structure. A similar mechanism is invoked to explain why noise increases the robustness of the STO regime. Taking advantage of the reduction of the dimensionality of the full stellate cell system, we propose a nonlinear artificially spiking (NAS) model in which the STI reduced system is supplemented with a threshold for spiking and a reset voltage. We show that the synchronization properties in networks made up of the NAS cells are similar to those of networks using the full stellate cell models. In memory of Angel A. Alonso     

Stability switches,oscillatory multistability,and spatio-temporal patterns of nonlinear oscillations in recurrently delay coupled neural networks

A model of time-delay recurrently coupled spatially segregated neural assemblies is here proposed. We show that it operates like some of the hierarchical architectures of the brain. Each assembly is a neural network with no delay in the local couplings between the units. The delay appears in the long range feedforward and feedback inter-assemblies communications. Bifurcation analysis of a simple four-units system in the autonomous case shows the richness of the dynamical behaviors in a biophysically plausible parameter region. We find oscillatory multistability, hysteresis, and stability switches of the rest state provoked by the time delay. Then we investigate the spatio-temporal patterns of bifurcating periodic solutions by using the symmetric local Hopf bifurcation theory of delay differential equations and derive the equation describing the flow on the center manifold that enables us determining the direction of Hopf bifurcations and stability of the bifurcating periodic orbits. We also discuss computational properties of the system due to the delay when an external drive of the network mimicks external sensory input.     

Inhibition of rhythmic spiking by colored noise in neural systems

We study the effect of colored noise on the rhythmic spiking activity of neural systems in this paper. The phenomenon of the so-called inverse stochastic resonance , that is, noise with appropriate intensity suppresses the spiking activity in neural systems, is clearly observed in a special parameter regime. We find that the inhibition effect of colored noise is stronger than that of Gaussian white noise. Furthermore, our simulation results show that the inhibition effect of colored noise provides a useful mechanism for the generation of synchronized burst in type-2 mixed-feed-forward-feedback loop neuronal network motif, which indicates that such inhibition effect might have some biological implications.