Some conditions for sustained oscillations in chain processes with feedback inhibition and saturable removal

A biochemical chain with feedback inhibition, enzymatic removal and catalyzed input is considered. Some conditions on the parameters are given for which the result is a simple limit cycle behavior or a limit cycle surrounding an unstable limit cycle and a stable point. In the latter case, the system in the steady state can either be at rest or oscillate around the point at rest, depending on the initial conditions. Presented at the May 23–24, 1975 meeting of the Society for Mathematical Biology, Bowling Green State University, Bowling Green, Ohio.     

Mathematical biophysics of sustained nonperiodic oscillations of excitation and inhibition in the central nervous system

There are a number of phenomena in the central nervous system which are not periodic in the strict sense of the word. The simplest example is the α-rhythm which represents an oscillation that is not strictly periodic in a mathematical sense. Some diseases like periodic relapsing catatonia or the manic-depressive psychoses are almost strictly periodical but such phenomena as attacks of epilepsy repeat themselves perhaps at average intervals but those intervals are not equal and are not predictable. We consider that even such phenomena like a fit of creative work has the same mechanism. All those phenomena deal with sporadic spontaneous nonperiodic overexcitation of certain parts of the brain. Depending on which part of the brain is overexcited, we might have either a grand or a petit mal epilepsy, or a fit of creativity. This paper gives a model of two neuronic circuits which can produce nonperiodic spontaneous overexcitation of the cortex. That part of the cortex which happens to have the lowest threshold will be most strongly affected. The location of such parts of the cortex vary from person to person. The paper does not in any way imply that the mechanism described here is responsible for all types of epilepsy. Many of them may have an entirely different origin. The paper does, however, contain a suggestion for an actual cure of some types of epilepsy, which is different from the symptomatic treatment mostly used so far.     

Membrane properties and the balance between excitation and inhibition control gamma-frequency oscillations arising from feedback inhibition

Computational studies as well as in vivo and in vitro results have shown that many cortical neurons fire in a highly irregular manner and at low average firing rates. These patterns seem to persist even when highly rhythmic signals are recorded by local field potential electrodes or other methods that quantify the summed behavior of a local population. Models of the 30-80 Hz gamma rhythm in which network oscillations arise through 'stochastic synchrony' capture the variability observed in the spike output of single cells while preserving network-level organization. We extend upon these results by constructing model networks constrained by experimental measurements and using them to probe the effect of biophysical parameters on network-level activity. We find in simulations that gamma-frequency oscillations are enabled by a high level of incoherent synaptic conductance input, similar to the barrage of noisy synaptic input that cortical neurons have been shown to receive in vivo. This incoherent synaptic input increases the emergent network frequency by shortening the time scale of the membrane in excitatory neurons and by reducing the temporal separation between excitation and inhibition due to decreased spike latency in inhibitory neurons. These mechanisms are demonstrated in simulations and in vitro current-clamp and dynamic-clamp experiments. Simulation results further indicate that the membrane potential noise amplitude has a large impact on network frequency and that the balance between excitatory and inhibitory currents controls network stability and sensitivity to external inputs.     

Perisomatic feedback inhibition underlies cholinergically induced fast network oscillations in the rat hippocampus in vitro

Gamma frequency network oscillations are assumed to be important in cognitive processes, including hippocampal memory operations, but the precise functions of these oscillations remain unknown. Here, we examine the cellular and network mechanisms underlying carbachol-induced fast network oscillations in the hippocampus in vitro, which closely resemble hippocampal gamma oscillations in the behaving rat. Using a combination of planar multielectrode array recordings, imaging with voltage-sensitive dyes, and recordings from single hippocampal neurons within the CA3 gamma generator, active current sinks and sources were localized to the stratum pyramidale. These proximal currents were driven by phase-locked rhythmic inhibitory inputs to pyramidal cells from identified perisomatic-targeting interneurons. AMPA receptor-mediated recurrent excitation was necessary for the synchronization of interneuronal discharge, which strongly supports a synaptic feedback model for the generation of hippocampal gamma oscillations.     

Computer simulation of sustained oscillations in peroxidase-oxidase reaction

A system of differential equations of second order exhibiting transitional behaviour and sustained oscillations has been obtained for a complete scheme of the peroxidase-oxidase reaction. The concentrations of hydrogen peroxide and of hydrogen donor radicals are slow variables of the system. The most essential reactions responsible for oscillations have been selected. Analysis of the system in phase plane and in parameter space has been carried out. The dependence of oscillation period and amplitude on the parameter values has been investigated.     

Optimizing interneuron circuits for compartment-specific feedback inhibition

Cortical circuits process information by rich recurrent interactions between excitatory neurons and inhibitory interneurons. One of the prime functions of interneurons is to stabilize the circuit by feedback inhibition, but the level of specificity on which inhibitory feedback operates is not fully resolved. We hypothesized that inhibitory circuits could enable separate feedback control loops for different synaptic input streams, by means of specific feedback inhibition to different neuronal compartments. To investigate this hypothesis, we adopted an optimization approach. Leveraging recent advances in training spiking network models, we optimized the connectivity and short-term plasticity of interneuron circuits for compartment-specific feedback inhibition onto pyramidal neurons. Over the course of the optimization, the interneurons diversified into two classes that resembled parvalbumin (PV) and somatostatin (SST) expressing interneurons. Using simulations and mathematical analyses, we show that the resulting circuit can be understood as a neural decoder that inverts the nonlinear biophysical computations performed within the pyramidal cells. Our model provides a proof of concept for studying structure-function relations in cortical circuits by a combination of gradient-based optimization and biologically plausible phenomenological models.     

Sporadic sustained nonperiodic oscillations in the activities of organismic sets

In combining the author's theories of organismic sets (Rashevsky,Bull. Math. Biophysics,31, 159–198, 1969a) and Robert Rosen's theory of (M, R)-systems (Bull. Math. Biophysics,20, 245–265, 1958), a conclusion is reached that the number of either normal or pathological phenomena in organismic sets may occur. Those phenomena are characterized by occurring spontaneously once in a while but are not exactly periodic. Some epilepsies are an example of such pathological phenomena in the brain.     

A new kinetic model for biochemical oscillations: Graph-theoretical analysis

A graphical analysis demonstrates the ability of slow substrate activation and certain types of cooperativity between the two enzyme active sites to generate sustained oscillations. The analysis allows us to estimate kinetic parameter values for which oscillations exist. The scheme analyzed can explain the cyclical changes in functioning of various motor enzymes. Moreover, this scheme does not generate bistability for any parameter values. The graphical analysis presented is simple and visually clarifies the regulatory role of the details in the kinetic schemes.     

Optimization of a stagewise biochemical reactor system with product feedback

A consecutive, first-order, irreversible, biochemical reaction, \documentclass{article}\pagestyle{empty}\begin{document}$ A{\textstyle{{k(\theta)} \over {{\rm Enzyme }1}}} \to B{\textstyle{{k(\theta)} \over {{\rm Enzyme 2}}}} \to C $\end{document}, taking place in a series of N reactors with product recycle is considered. A discrete version of the maximum principle is used to derive general equations necessary for maximizing the production of (1) the final product, C, by choosing the temperature or the pH value in each reactor, and (2) the intermediate product, B, by choosing the reactor volume. A numerical computation for a series of three reactors with recycle is illustrated. The effects of varying the recycle rates on the optimal state and decision variables are also presented.     

Spiral feedback oscillations of growing hypocotyl with radicle inPisum sativum L.

Cinematographic records of longitudinal growth showed that hypocotyl with radicle inPisum sativum L. undergoes spiral oscillations during growth. This phenomenon can be characterized by the following time-space limits:

1. (1)
   Curvature of the hypocotyl with radicle takes place always in the zone of most rapid elongation (Fig. 2, 8).     
   
   

A role for MAPK in feedback inhibition of Tcrb recombination

The Tcrb locus is subject to a host of regulatory mechanisms that impart a strict cell and developmental stage-specific order to variable (V), diversity (D), and joining (J) gene segment recombination. The Tcrb locus is also regulated by allelic exclusion mechanisms, which restrict functional rearrangements to a single allele. The production of a functional rearrangement in CD4-CD8- double-negative (DN) thymocytes leads to the assembly of a pre-TCR and initiates signaling cascades that allow for DN to CD4+CD8+ double-positive (DP) differentiation, proliferation, and feedback inhibition of further Vbeta to DJbeta rearrangement. Feedback inhibition is believed to be controlled, in part, by the loss of Vbeta gene segment accessibility during the DN to DP transition. However, the pre-TCR signaling pathways that lead to the inactivation of Vbeta chromatin have not been determined. Because activation of the MAPK pathway is documented to promote DP differentiation in the absence of allelic exclusion, we characterized the properties of Vbeta chromatin within DP thymocytes generated by a constitutively active Raf1 (Raf-CAAX) transgene. Consistent with previous reports, we show that the Raf-CAAX transgene does not inhibit Tcrb recombination in DN thymocytes. Nevertheless, DP thymocytes generated by Raf-CAAX signals display normal down-regulation of Vbeta segment accessibility and normal feedback inhibition of the Vbeta to DJbeta rearrangement. Therefore, our results emphasize the distinct requirements for feedback inhibition in the DN and DP compartments. Although MAPK activation cannot impose feedback in DN thymocytes, it contributes to feedback inhibition through developmental changes that are tightly linked to DN to DP differentiation.     

Slow oscillations in blood pressure via a nonlinear feedback model

Blood pressure is well established to contain a potential oscillation between 0.1 and 0.4 Hz, which is proposed to reflect resonant feedback in the baroreflex loop. A linear feedback model, comprising delay and lag terms for the vasculature, and a linear proportional derivative controller have been proposed to account for the 0.4-Hz oscillation in blood pressure in rats. However, although this model can produce oscillations at the required frequency, some strict relationships between the controller and vasculature parameters must be true for the oscillations to be stable. We developed a nonlinear model, containing an amplitude-limiting nonlinearity that allows for similar oscillations under a very mild set of assumptions. Models constructed from arterial pressure and sympathetic nerve activity recordings obtained from conscious rabbits under resting conditions suggest that the nonlinearity in the feedback loop is not contained within the vasculature, but rather is confined to the central nervous system. The advantage of the model is that it provides for sustained stable oscillations under a wide variety of situations even where gain at various points along the feedback loop may be altered, a situation that is not possible with a linear feedback model. Our model shows how variations in some of the nonlinearity characteristics can account for growth or decay in the oscillations and situations where the oscillations can disappear altogether. Such variations are shown to accord well with observed experimental data. Additionally, using a nonlinear feedback model, it is straightforward to show that the variation in frequency of the oscillations in blood pressure in rats (0.4 Hz), rabbits (0.3 Hz), and humans (0.1 Hz) is primarily due to scaling effects of conduction times between species.     

Some system considerations of neuron pools with feedback

This paper describes an approach to the analysis of the inputoutput relationships present in a neuron pool that receives a number of inputs. These inputs consist of primary inputs to the neuron pool and inputs resulting from feedback of information from the neuron pool as well. Multiple input-output relationships are obtained in terms of the synaptic weightings of the inputs, the membrane response characteristics of the neurons and the conduction delays on the feedback pathways.y=(I+MHD) ^-1 ·MFx is the explicit representation of the cell pool behavior assuming quasi-linear conditions, wherey is the output vector of cell responses,I is the identity matrix,M is the response matrix of the cells,H is the feedback synaptic weighting matrix,D is the delay matrix,F is the input weighting matrix, andx is the input vector. It is shown that a solution to this formulation exists, is unique, is stable, and can be computed by specified algorithms. An insight gained from this formulation suggests that the output of each cell in the pool is related to virtually all of the inputs to the pool and the outputs of all cells in the pool.Supported by NSF Grant GK 38301 and US-PHS Grant NS 08470.Program in Bioengineering, Electrical and Computer Engineering Department, The University of Michigan.Computer Information and Control Program, and Department of Electrical and Computer Engineering, The University of Michigan.     

S6K1 regulates GSK3 under conditions of mTOR-dependent feedback inhibition of Akt

Feedback inhibition of the PI3K-Akt pathway by the mammalian target of rapamycin complex 1 (mTORC1) has emerged as an important signaling event in tumor syndromes, cancer, and insulin resistance. Cells lacking the tuberous sclerosis complex (TSC) gene products are a model for this feedback regulation. We find that, despite Akt attenuation, the Akt substrate GSK3 is constitutively phosphorylated in cells and tumors lacking TSC1 or TSC2. In these settings, GSK3 phosphorylation is sensitive to mTORC1 inhibition by rapamycin or amino acid withdrawal, and GSK3 becomes a direct target of S6K1. This aberrant phosphorylation leads to decreased GSK3 activity and phosphorylation of downstream substrates and contributes to the growth-factor-independent proliferation of TSC-deficient cells. We find that GSK3 can also be regulated downstream of mTORC1 in a HepG2 model of cellular insulin resistance. Therefore, we define conditions in which S6K1, rather than Akt, is the predominant GSK3 regulatory kinase.