Instability in a generalized Keller–Segel model

We present a generalized Keller–Segel model where an arbitrary number of chemical compounds react, some of which are produced by a species, and one of which is a chemoattractant for the species. To investigate the stability of homogeneous stationary states of this generalized model, we consider the eigenvalues of a linearized system. We are able to reduce this infinite dimensional eigenproblem to a parametrized finite dimensional eigenproblem. By matrix theoretic tools, we then provide easily verifiable sufficient conditions for destabilizing the homogeneous stationary states. In particular, one of the sufficient conditions is that the chemotactic feedback is sufficiently strong. Although this mechanism was already known to exist in the original Keller–Segel model, here we show that it is more generally applicable by significantly enlarging the class of models exhibiting this instability phenomenon which may lead to pattern formation.     

The complex dynamics of a diffusive prey–predator model with an Allee effect in prey

This paper investigates complex dynamics of a predator–prey interaction model that incorporates: (a) an Allee effect in prey; (b) the Michaelis–Menten type functional response between prey and predator; and (c) diffusion in both prey and predator. We provide rigorous mathematical results of the proposed model including: (1) the stability of non-negative constant steady states; (2) sufficient conditions that lead to Hopf/Turing bifurcations; (3) a prior estimates of positive steady states; (4) the non-existence and existence of non-constant positive steady states when the model is under zero-flux boundary condition. We also perform completed analysis of the corresponding ODE model to obtain a better understanding on effects of diffusion on the stability. Our analytical results show that the small values of the ratio of the prey's diffusion rate to the predator's diffusion rate are more likely to destabilize the system, thus generate Hopf-bifurcation and Turing instability that can lead to different spatial patterns. Through numerical simulations, we observe that our model, with or without Allee effect, can exhibit extremely rich pattern formations that include but not limit to strips, spotted patterns, symmetric patterns. In addition, the strength of Allee effects also plays an important role in generating distinct spatial patterns.     

Studies on chlorophyll biosynthesis and other things

Our research on chlorophyll biosynthesis, over a period of approximately twenty years, has been described, emphasizing those areas in which our laboratory made significant and timely contributions. References to some of our most important articles are included. Portions of the chlorophyll biosynthetic pathway, in which our own laboratory was not involved, for example, the reduction of protochlorophyllide to chlorophyllide and the phytylation of the latter to yield chlorophyll a, have not been covered in this article. Those events which preceded my involvement with chlorophyll biosynthesis, but which contributed to the formation of my own scientific personality, are mentioned briefly in the Introduction. My non-scientific avocations have been included at the request of the reviewers and Govindjee.     

Self-organization and productivity in semi-arid ecosystems: implications of seasonality in rainfall

Spatial self-organization including striking vegetation patterns observed in arid ecosystems has been studied in models with uniform rainfall. In this paper, we present a fully seasonal rainfall model that produces vegetation patterns found in nature by including the natural adaptation of plants to scarcity of water and the consequent seasonal variation in their growth and metabolic rate. We present results for the mean-field and spatially extended versions of the model. We find that the patterns depend on the duration of the wet season even with fixed total annual precipitation (PPT) showing how seasonality affects spatial self-organization. We observe that the productivity can vary for fixed PPT as a function of the duration thereby providing another source of observed variations. We compute the maximum vegetation cover as function of PPT and find that the behavior is consistent with observations. We comment on the implications for regime shifts due to increased interannual fluctuations caused by climatic changes. Our specific model calculations provide more general conclusions for ecosystems with competition for scarce resources due to seasonal variations in the resource, especially for self-organization and productivity.     

Do rats time filled and empty intervals of equal duration differently?

The goal was to determine whether rats time filled and empty intervals of equal duration differently. Each of five rats was trained for 50 sessions on an instrumental appetitive head entry procedure in which food was available (primed) every 120 s. On "empty" cycles, 30s prior to the next food prime, a 0.5-s pulse of white noise was presented. On "filled" cycles, 30s prior to the next food prime, white noise came on and stayed on until food was delivered. The two types of cycles were presented with equal probability. The results showed that the rats timed both the food-to-food interval and the stimulus-to-food interval. A comparison of the response gradients on filled and empty cycles following stimulus presentation showed better temporal discrimination on filled cycles. The results were modeled using a Packet theory of timing, with a linear averaging rule to combine the temporal information provided by the stimulus and food. The model fits to the individual response gradients were evaluated with a Turing test.     

Gap junctions modulate seizures in a mean-field model of general anesthesia for the cortex

During slow-wave sleep, general anesthesia, and generalized seizures, there is an absence of consciousness. These states are characterized by low-frequency large-amplitude traveling waves in scalp electroencephalogram. Therefore the oscillatory state might be an indication of failure to form coherent neuronal assemblies necessary for consciousness. A generalized seizure event is a pathological brain state that is the clearest manifestation of waves of synchronized neuronal activity. Since gap junctions provide a direct electrical connection between adjoining neurons, thus enhancing synchronous behavior, reducing gap-junction conductance should suppress seizures; however there is no clear experimental evidence for this. Here we report theoretical predictions for a physiologically-based cortical model that describes the general anesthetic phase transition from consciousness to coma, and includes both chemical synaptic and direct electrotonic synapses. The model dynamics exhibits both Hopf (temporal) and Turing (spatial) instabilities; the Hopf instability corresponds to the slow (≲8 Hz) oscillatory states similar to those seen in slow-wave sleep, general anesthesia, and seizures. We argue that a delicately balanced interplay between Hopf and Turing modes provides a canonical mechanism for the default non-cognitive rest state of the brain. We show that the Turing mode, set by gap-junction diffusion, is generally protective against entering oscillatory modes; and that weakening the Turing mode by reducing gap conduction can release an uncontrolled Hopf oscillation and hence an increased propensity for seizure and simultaneously an increased sensitivity to GABAergic anesthesia.     

Elevated plasma vasopressin in cardiomyopathic hamsters

Hamsters of the BIO 14.6 strain characteristically develop cardiomyopathy as they age, and hamsters of this strain have overt signs of heart failure by 11 months of age. Plasma levels of the posterior pituitary hormone arginine-vasopressin (AVP) were found to be elevated (approximately 2-fold) in 11 month old BIO 14.6 hamsters, compared to age-matched hamsters of a control strain. AVP appeared inappropriately elevated in these animals, since they were neither hyperosmotic nor markedly hypotensive. The elevated levels of AVP observed in these animals appears to contribute to vasomotor tone, since intravenous adminstration of a specific antagonist of the vasoconstrictor action of AVP [d(CH₂)₅Ome(TYR)AVP] elicited a fall in arterial pressure (9±2 mm Hg, n=6, p<0.05). The AVP antagonist had no effect on arterial pressure in hamsters of a control strain, and vehicle administration had no effect on arterial pressure in either strain. These data indicate that inappropriately elevated levels of AVP contribute to the cardiovascular state of myopathic hamsters. Since elevated plasma AVP has been noted in human congestive heart failure, these results suggest that AVP may contribute to the cardiovascular status during congestive heart failure.     

Perspective: network-guided pattern formation of neural dynamics

The understanding of neural activity patterns is fundamentally linked to an understanding of how the brain''s network architecture shapes dynamical processes. Established approaches rely mostly on deviations of a given network from certain classes of random graphs. Hypotheses about the supposed role of prominent topological features (for instance, the roles of modularity, network motifs or hierarchical network organization) are derived from these deviations. An alternative strategy could be to study deviations of network architectures from regular graphs (rings and lattices) and consider the implications of such deviations for self-organized dynamic patterns on the network. Following this strategy, we draw on the theory of spatio-temporal pattern formation and propose a novel perspective for analysing dynamics on networks, by evaluating how the self-organized dynamics are confined by network architecture to a small set of permissible collective states. In particular, we discuss the role of prominent topological features of brain connectivity, such as hubs, modules and hierarchy, in shaping activity patterns. We illustrate the notion of network-guided pattern formation with numerical simulations and outline how it can facilitate the understanding of neural dynamics.