Mechanobiology in cortical waves and oscillations

Cortical actin waves have emerged as a widely prevalent phenomena and brought pattern formation to many fields of cell biology. Cortical excitabilities, reminiscent of the electric excitability in neurons, are likely fundamental property of the cell cortex. Although they have been mostly considered to be biochemical in nature, accumulating evidence support the role of mechanics in the pattern formation process. Both pattern formation and mechanobiology approach biological phenomena at the collective level, either by looking at the mesoscale dynamical behavior of molecular networks or by using collective physical properties to characterize biological systems. As such they are very different from the traditional reductionist, bottom-up view of biology, which brings new challenges and potential opportunities. In this essay, we aim to provide our perspectives on what the proposed mechanochemical feedbacks are and open questions regarding their role in cortical excitable and oscillatory dynamics.     

Calcium waves and oscillations in eggs

Eggs from several protostomes (molluscs, annelids, nemerteans, etc.) and two deuterostomes (mammals and ascidians) display repetitive calcium signals. Oscillations in the level of intracellular calcium concentration are occasionally triggered by maturing hormones (as in some molluscs) and mostly observed after fertilization which occurs at different stages of the meiotic cell cycle (oocytes are arrested in prophase, metaphase I or metaphase II). In most eggs examined so far, calcium oscillations last until the end of meiosis just before male and female pronuclei form. This ability depends on the sensitivity of InsP3 channels and on the permeability of the plasma membrane to extracellular calcium. In eggs that undergo cytoplasmic reorganization at fertilization (annelids, nemerteans, ascidians, etc.) the repetitive calcium signals are waves that originate from localized cortical sites that become calcium waves pacemakers. In ascidians we have identified the site of initiation of repetitive calcium waves as an accumulation of endoplasmic reticulum sandwiched between the plasma membrane and an accumulation of mitochondria. We compare and discuss the generation of calcium signals in the different eggs, their relationship with the cell cycle and the possible roles they play during development.     

Working memory dynamics and spontaneous activity in a flip-flop oscillations network model with a Milnor attractor

Many cognitive tasks require the ability to maintain and manipulate simultaneously several chunks of information. Numerous neurobiological observations have reported that this ability, known as the working memory, is associated with both a slow oscillation (leading to the up and down states) and the presence of the theta rhythm. Furthermore, during resting state, the spontaneous activity of the cortex exhibits exquisite spatiotemporal patterns sharing similar features with the ones observed during specific memory tasks. Here to enlighten neural implication of working memory under these complicated dynamics, we propose a phenomenological network model with biologically plausible neural dynamics and recurrent connections. Each unit embeds an internal oscillation at the theta rhythm which can be triggered during up-state of the membrane potential. As a result, the resting state of a single unit is no longer a classical fixed point attractor but rather the Milnor attractor, and multiple oscillations appear in the dynamics of a coupled system. In conclusion, the interplay between the up and down states and theta rhythm endows high potential in working memory operation associated with complexity in spontaneous activities.

Linking neural activity and molecular oscillations in the SCN

Neurons in the suprachiasmatic nucleus (SCN) function as part of a central timing circuit that drives daily changes in our behaviour and underlying physiology. A hallmark feature of SCN neuronal populations is that they are mostly electrically silent during the night, start to fire action potentials near dawn and then continue to generate action potentials with a slow and steady pace all day long. Sets of currents are responsible for this daily rhythm, with the strongest evidence for persistent Na(+) currents, L-type Ca(2+) currents, hyperpolarization-activated currents (I(H)), large-conductance Ca(2+) activated K(+) (BK) currents and fast delayed rectifier (FDR) K(+) currents. These rhythms in electrical activity are crucial for the function of the circadian timing system, including the expression of clock genes, and decline with ageing and disease. This article reviews our current understanding of the ionic and molecular mechanisms that drive the rhythmic firing patterns in the SCN.     

Linking the mitochondrial genotype to the organismal phenotype

One of the grand challenges of the postgenomics era is to mechanistically link the genotype with the phenotype. Here, we consider the link between the mitochondrial genotype and the organismal phenotype that is provided by bioenergetic studies of the electron transport chain. That linkage is pertinent for the fields of molecular ecology and phylogeography as it tests if, and potentially how, natural selection can influence the evolutionary and demographic past of both populations and species. We introduce the mitochondrial genotype in terms of mitochondrial‐encoded genes, nuclear‐encoded genes that produce structural proteins imported into the mitochondria, and mitochondrial DNA–nuclear interactions. We then review the potential for quaternary structure modelling to predict the functional consequence of specific naturally occurring mutations. We discuss how the energy‐producing reactions of oxidative phosphorylation can be used to provide a mechanistic biochemical link between genotype and phenotype. Experimental manipulations can then be used to test the functional consequences of specific mutations in multiple genetic backgrounds. Finally, we examine how mitochondria can influence the organismal mitochondrial phenotype using the examples of lifespan, fertility and starvation resistance and discuss how mitochondria may be involved in establishing both the upper and lower thermal limits of organisms. We conclude that mitochondrial DNA mutations can be important in determining aspects of organism life history. The question that remains to be resolved is how common are these adaptive mutations?     

The binding problem in population neurodynamics: a network model for stimulus-specific coherent oscillations

A hypothesis is presented that coherent oscillatory discharges of spatially distributed neuronal groups (the supposed binding mechanism) are the result of the convergence of stimulus-dependent activity in modality-specific afferent pathways with oscillatory activity generated in unspecific sensory systems. This view is supported by simulation experiments on model networks.     

Forced oscillations in a windkessel model

Volume elasticity of the arterial system and its component parts is developed starting from a Windkessel^*-model, which is defined in 4 points. Emphasis is laid on the simplicity of the derived equations and accessibility to experimental verification. The theory is an extension of earlier work achieved by Wetterer and Pieper (1953), who introduced an essentially physical method for the indirect determination of volume elasticity in situ, by creating forced sinusoidal oscillations in the arterial system, using a special pump operated at a considerably lower frequency than the mean heart frequency. The elegance of both experimental technique and the derived equations incited us to investigate the mathematical foundation and possible generalization of the method.     

Forced oscillations in a windkessel model

Volume elasticity of the arterial system and its component parts is developed starting from a Windkessel^*-model, which is defined in 4 points. Emphasis is laid on the simplicity of the derived equations and accessibility to experimental verification. The theory is an extension of earlier work achieved by Wetterer and Pieper (1953), who introduced an essentially physical method for the indirect determination of volume elasticity in situ, by creating forced sinusoidal oscillations in the arterial system, using a special pump operated at a considerably lower frequency than the mean heart frequency. The elegance of both experimental technique and the derived equations incited us to investigate the mathematical foundation and possible generalization of the method.     

A single-pool model for intracellular calcium oscillations and waves in the Xenopus laevis oocyte.

We construct a minimal model of cytosolic free Ca2+ oscillations based on Ca2+ release via the inositol 1,4,5-trisphosphate (IP3) receptor/Ca2+ channel (IP3R) of a single intracellular Ca2+ pool. The model relies on experimental evidence that the cytosolic free calcium concentration ([Ca2+]c) modulates the IP3R in a biphasic manner, with Ca2+ release inhibited by low and high [Ca2+]c and facilitated by intermediate [Ca2+]c, and that channel inactivation occurs on a slower time scale than activation. The model produces [Ca2+]c oscillations at constant [IP3] and reproduces a number of crucial experiments. The two-dimensional spatial model with IP3 dynamics, cytosolic diffusion of IP3 (Dp = 300 microns 2 s-1), and cytosolic diffusion of Ca2+ (Dc = 20 microns 2 s-1) produces circular, planar, and spiral waves of Ca2+ with speeds of 7-15 microns.s-1, which annihilate upon collision. Increasing extracellular [Ca2+] influx increases wave speed and baseline [Ca2+]c. A [Ca2+]c-dependent Ca2+ diffusion coefficient does not alter the qualitative behavior of the model. An important model prediction is that channel inactivation must occur on a slower time scale than activation in order for waves to propagate. The model serves to capture the essential macroscopic mechanisms that are involved in the production of intracellular Ca2+ oscillations and traveling waves in the Xenopus laevis oocyte.     

Linking a spatially-explicit model of acacias to GIS and remotely-sensed data

Spatially-explicit and landscape-related simulation models are increasingly used in ecology, but are often criticized because their parameterization has high data requirements. A frequently suggested approach to overcome this difficulty is the linkage of spatially-explicit or landscape-related models with GIS (geographic information system) and remote-sensing technology. GIS can provide data on relevant landscape features, such as topography, and satellite images can be used to identify spatial vegetation distribution. In this paper, we use these techniques for simple, cost-inexpensive (in both time and money) parameterization based on readily-available GIS and remotely-sensed data. We use a previously developed, spatially-explicit model of the population dynamics of anAcacia species in the Negev desert of Israel (SAM, spatialAcacia model) to investigate if model initialization (measurement of current tree distribution) can be obtained from readily-available satellite images using a radiometric vegetation index (NDVI, normalized difference vegetation index). Furthermore, we investigate the applicability and the advantages of using an explicit consideration of landscape features in the model based on topographic data from a GIS. Using a DEM (digital elevation model), we compare the wadi topography to the current tree distribution observed in the field.     

On a recurrent neural network producing oscillations

A recurrent two-node neural network producing oscillations is analyzed. The network has no true inputs and the outputs from the network exhibit a circular phase portrait. The weight configuration of the network is investigated, resulting in analytical weight expressions, which are compared with numerical weight estimates obtained by training the network on the desired trajectories. The values predicted by the analytical expressions agree well with the findings from the numerical study, and can also explain the asymptotic properties of the networks studied.     

The generation of phase differences and frequency changes in a network model of inferior olive subthreshold oscillations

It is commonly accepted that the Inferior Olive (IO) provides a timing signal to the cerebellum. Stable subthreshold oscillations in the IO can facilitate accurate timing by phase-locking spikes to the peaks of the oscillation. Several theoretical models accounting for the synchronized subthreshold oscillations have been proposed, however, two experimental observations remain an enigma. The first is the observation of frequent alterations in the frequency of the oscillations. The second is the observation of constant phase differences between simultaneously recorded neurons. In order to account for these two observations we constructed a canonical network model based on anatomical and physiological data from the IO. The constructed network is characterized by clustering of neurons with similar conductance densities, and by electrical coupling between neurons. Neurons inside a cluster are densely connected with weak strengths, while neurons belonging to different clusters are sparsely connected with stronger connections. We found that this type of network can robustly display stable subthreshold oscillations. The overall frequency of the network changes with the strength of the inter-cluster connections, and phase differences occur between neurons of different clusters. Moreover, the phase differences provide a mechanistic explanation for the experimentally observed propagating waves of activity in the IO. We conclude that the architecture of the network of electrically coupled neurons in combination with modulation of the inter-cluster coupling strengths can account for the experimentally observed frequency changes and the phase differences.     

Linking species functional roles to their network roles

Species roles in ecological networks combine to generate their architecture, which contributes to their stability. Species trait diversity also affects ecosystem functioning and resilience, yet it remains unknown whether species’ contributions to functional diversity relate to their network roles. Here, we use 21 empirical pollen transport networks to characterise this relationship. We found that, apart from a few abundant species, pollinators with original traits either had few interaction partners or interacted most frequently with a subset of these partners. This suggests that narrowing of interactions to a subset of the plant community accompanies pollinator niche specialisation, congruent with our hypothesised trade‐off between having unique traits vs. being able to interact with many mutualist partners. Conversely, these effects were not detected in plants, potentially because key aspects of their flowering traits are conserved at a family level. Relating functional and network roles can provide further insight into mechanisms underlying ecosystem functioning.     

Transitions to spike-wave oscillations and epileptic dynamics in a human cortico-thalamic mean-field model

In this paper we present a detailed theoretical analysis of the onset of spike-wave activity in a model of human electroencephalogram (EEG) activity, relating this to clinical recordings from patients with absence seizures. We present a complete explanation of the transition from inter-ictal activity to spike and wave using a combination of bifurcation theory, numerical continuation and techniques for detecting the occurrence of inflection points in systems of delay differential equations (DDEs). We demonstrate that the initial transition to oscillatory behaviour occurs as a result of a Hopf bifurcation, whereas the addition of spikes arises as a result of an inflection point of the vector field. Strikingly these findings are consistent with EEG data recorded from patients with absence seizures and we present a discussion of the clinical significance of these results, suggesting potential new techniques for detection and anticipation of seizures.     

Backward bifurcation and oscillations in a nested immuno-eco-epidemiological model

This paper introduces a novel partial differential equation immuno-eco-epidemiological model of competition in which one species is affected by a disease while another can compete with it directly and by lowering the first species' immune response to the infection, a mode of competition termed stress-induced competition. When the disease is chronic, and the within-host dynamics are rapid, we reduce the partial differential equation model (PDE) to a three-dimensional ordinary differential equation (ODE) model. The ODE model exhibits backward bifurcation and sustained oscillations caused by the stress-induced competition. Furthermore, the ODE model, although not a special case of the PDE model, is useful for detecting backward bifurcation and oscillations in the PDE model. Backward bifurcation related to stress-induced competition allows the second species to persist for values of its invasion number below one. Furthermore, stress-induced competition leads to destabilization of the coexistence equilibrium and sustained oscillations in the PDE model. We suggest that complex systems such as this one may be studied by appropriately designed simple ODE models.     

Canards and mixed-mode oscillations in a forest pest model

We consider a three-variable forest pest model, proposed by Rinaldi & Muratori (1992) [Rinaldi, S., Muratori, S., 1992. Limit cycles in slow–fast forest–pest models. Theor. Popul. Biol. 41, 26–43]. The model allows relaxation oscillations where long pest-free periods are interspersed with outbreaks of high pest concentration. For small values of the timescale of the young trees, the model can be reduced to a two-dimensional model. By a geometrical analysis we identify a canard explosion in the reduced model, that is, a change over a narrow parameter interval from outbreak dynamics to small oscillations around an endemic state. For larger values of the timescale of the young trees the two-dimensional approximation breaks down, and a broader parameter interval with mixed-mode oscillations appear, replacing the simple canard explosion. The analysis only relies on simple and generic properties of the model, and is expected to be applicable in a larger class of multiple timescale dynamical models.     

Multiple-spike waves in a one-dimensional integrate-and-fire neural network

This paper builds on the past study of single-spike waves in one-dimensional integrate-and-fire networks to provide a framework for the study of waves with arbitrary (finite or countably infinite) collections of spike times. Based on this framework, we prove an existence theorem for single-spike traveling waves, and we combine analysis and numerics to study two-spike traveling waves, periodic traveling waves, and general infinite spike trains. For a fixed wave speed, finite-spike waves, periodic waves, and other infinite-spike waves may all occur, and we discuss the relationships among them. We also relate the waves considered analytically to waves generated in numerical simulations by the transient application of localized excitation.Key words or phrases:Traveling waves, Integrate-and-fire network, Excitatory synaptic coupling