Integrate and fire neural networks, piecewise contractive maps and limit cycles

We study the global dynamics of integrate and fire neural networks composed of an arbitrary number of identical neurons interacting by inhibition and excitation. We prove that if the interactions are strong enough, then the support of the stable asymptotic dynamics consists of limit cycles. We also find sufficient conditions for the synchronization of networks containing excitatory neurons. The proofs are based on the analysis of the equivalent dynamics of a piecewise continuous Poincaré map associated to the system. We show that for efficient interactions the Poincaré map is piecewise contractive. Using this contraction property, we prove that there exist a countable number of limit cycles attracting all the orbits dropping into the stable subset of the phase space. This result applies not only to the Poincaré map under study, but also to a wide class of general n-dimensional piecewise contractive maps.     

Emergent oscillations in networks of stochastic spiking neurons

Networks of neurons produce diverse patterns of oscillations, arising from the network's global properties, the propensity of individual neurons to oscillate, or a mixture of the two. Here we describe noisy limit cycles and quasi-cycles, two related mechanisms underlying emergent oscillations in neuronal networks whose individual components, stochastic spiking neurons, do not themselves oscillate. Both mechanisms are shown to produce gamma band oscillations at the population level while individual neurons fire at a rate much lower than the population frequency. Spike trains in a network undergoing noisy limit cycles display a preferred period which is not found in the case of quasi-cycles, due to the even faster decay of phase information in quasi-cycles. These oscillations persist in sparsely connected networks, and variation of the network's connectivity results in variation of the oscillation frequency. A network of such neurons behaves as a stochastic perturbation of the deterministic Wilson-Cowan equations, and the network undergoes noisy limit cycles or quasi-cycles depending on whether these have limit cycles or a weakly stable focus. These mechanisms provide a new perspective on the emergence of rhythmic firing in neural networks, showing the coexistence of population-level oscillations with very irregular individual spike trains in a simple and general framework.     

Cryptochrome-positive and -negative clock neurons in Drosophila entrain differentially to light and temperature

The blue-light photoreceptive protein Cryptochrome (CRY) plays an important role in the light synchronization of the Drosophila circadian clock. Previously, we found that among the approximately 150 clock neurons, many but not all neurons express CRY. We speculated that the CRY-positive pacemaker neurons may be especially important for light entrainment, whereas the CRY-negative neurons may be important for other environmental cues, for example, temperature. To investigate this hypothesis, we tested the entrainability of the clock neurons to out-of-phase light and temperature cycles. When light-dark or light-dim light cycles were shifted by 12 h with respect to temperature cycles, behavioral rhythms of wild-type flies were re-entrained by the light cycles. In this condition, we found that TIMELESS (TIM) level was strongly influenced by the temperature cycles in many CRY-negative clock neurons, suggesting that the CRY-negative neurons have higher sensitivity to temperature. Under the same conditions, cry-null mutants entrained to the temperature cycles or very slowly re-entrained to light-dark cycles. Our results suggest that there are 2 types of clock neurons having differential sensitivities to light and temperature, and CRY is a key component for the preferential entrainment to light.     

Variable neuronal participation in stereotypic motor programs

To what extent are motor networks underlying rhythmic behaviors rigidly hard-wired versus fluid and dynamic entities? Do the members of motor networks change from moment-to-moment or from motor program episode-to-episode? These are questions that can only be addressed in systems where it is possible to monitor the spiking activity of networks of neurons during the production of motor programs. We used large-scale voltage-sensitive dye (VSD) imaging followed by Independent Component Analysis spike-sorting to examine the extent to which the neuronal network underlying the escape swim behavior of Tritonia diomedea is hard-wired versus fluid from a moment-to-moment perspective. We found that while most neurons were dedicated to the swim network, a small but significant proportion of neurons participated in a surprisingly variable manner. These neurons joined the swim motor program late, left early, burst only on some cycles or skipped cycles of the motor program. We confirmed that this variable neuronal participation was not due to effects of the VSD by finding such neurons with intracellular recording in dye-free saline. Further, these neurons markedly varied their level of participation in the network from swim episode-to-episode. The generality of such unreliably bursting neurons was confirmed by their presence in the rhythmic escape networks of two other molluscan species, Tritonia festiva and Aplysia californica. Our observations support a view that neuronal networks, even those underlying rhythmic and stereotyped motor programs, may be more variable in structure than widely appreciated.     

Asymmetric response of early warning indicators of phytoplankton transition to and from cycles

Phytoplankton populations often exhibit cycles associated with nuisance blooms of cyanobacteria and other algae that cause toxicity, odor problems, oxygen depletion, and fish kills. Models of phytoplankton blooms used for management and basic research often contain critical transitions from stable points to cycles, or vice-versa. It would be useful to know whether aquatic systems, especially water supplies, are close to a critical threshold for cycling blooms. Recent studies of resilience indicators have focused on alternate stable points, although theory suggests that indicators such as variance and autocorrelation should also rise prior to a transition from stable point to stable cycle. We investigated changes in variance and autocorrelation associated with transitions involving cycles using two models. Variance rose prior to the transition from a small-radius cycle (or point) to a larger radius cycle in all cases. In many but not all cases, autocorrelation increased prior to the transition. However, the transition from large-radius to small-radius cycles was not associated with discernible increases in variance or autocorrelation. Thus, indicators of changing resilience can be measured prior to the transition from stable to cyclic plankton dynamics. Such indicators are potentially useful in management. However, these same indicators do not provide useful signals of the reverse transition, which is often a goal of aquatic ecosystem restoration. Thus, the availability of resilience indicators for phytoplankton cycles is asymmetric: the indicators are seen for the transition to bloom–bust cycles but not for the reverse transition to a phytoplankton stable point.     

On the stability of metabolic cycles

We investigate the stability properties of two different classes of metabolic cycles using a combination of analytical and computational methods. Using principles from structural kinetic modeling (SKM), we show that the stability of metabolic networks with certain structural regularities can be studied using a combination of analytical and computational techniques. We then apply these techniques to a class of single input, single output metabolic cycles, and find that the cycles are stable under all conditions tested. Next, we extend our analysis to a small autocatalytic cycle, and determine parameter regimes within which the cycle is very likely to be stable. We demonstrate that analytical methods can be used to understand the relationship between kinetic parameters and stability, and that results from these analytical methods can be confirmed with computational experiments. In addition, our results suggest that elevated metabolite concentrations and certain crucial saturation parameters can strongly affect the stability of the entire metabolic cycle. We discuss our results in light of the possibility that evolutionary forces may select for metabolic network topologies with a high intrinsic probability of being stable. Furthermore, our conclusions support the hypothesis that certain types of metabolic cycles may have played a role in the development of primitive metabolism despite the absence of regulatory mechanisms.     

Pavlovian fear conditioning activates a common pattern of neurons in the lateral amygdala of individual brains

Understanding the physical encoding of a memory (the engram) is a fundamental question in neuroscience. Although it has been established that the lateral amygdala is a key site for encoding associative fear memory, it is currently unclear whether the spatial distribution of neurons encoding a given memory is random or stable. Here we used spatial principal components analysis to quantify the topography of activated neurons, in a select region of the lateral amygdala, from rat brains encoding a Pavlovian conditioned fear memory. Our results demonstrate a stable, spatially patterned organization of amygdala neurons are activated during the formation of a Pavlovian conditioned fear memory. We suggest that this stable neuronal assembly constitutes a spatial dimension of the engram.     

Molecular switches at the synapse emerge from receptor and kinase traffic

Changes in the synaptic connection strengths between neurons are believed to play a role in memory formation. An important mechanism for changing synaptic strength is through movement of neurotransmitter receptors and regulatory proteins to and from the synapse. Several activity-triggered biochemical events control these movements. Here we use computer models to explore how these putative memory-related changes can be stabilised long after the initial trigger, and beyond the lifetime of synaptic molecules. We base our models on published biochemical data and experiments on the activity-dependent movement of a glutamate receptor, AMPAR, and a calcium-dependent kinase, CaMKII. We find that both of these molecules participate in distinct bistable switches. These simulated switches are effective for long periods despite molecular turnover and biochemical fluctuations arising from the small numbers of molecules in the synapse. The AMPAR switch arises from a novel self-recruitment process where the presence of sufficient receptors biases the receptor movement cycle to insert still more receptors into the synapse. The CaMKII switch arises from autophosphorylation of the kinase. The switches may function in a tightly coupled manner, or relatively independently. The latter case leads to multiple stable states of the synapse. We propose that similar self-recruitment cycles may be important for maintaining levels of many molecules that undergo regulated movement, and that these may lead to combinatorial possible stable states of systems like the synapse.     

Effects of conduction delays on the existence and stability of one to one phase locking between two pulse-coupled oscillators

Gamma oscillations can synchronize with near zero phase lag over multiple cortical regions and between hemispheres, and between two distal sites in hippocampal slices. How synchronization can take place over long distances in a stable manner is considered an open question. The phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike, depending upon where in the cycle it is received. We use PRCs under the assumption of pulsatile coupling to derive existence and stability criteria for 1:1 phase-locking that arises via bidirectional pulse coupling of two limit cycle oscillators with a conduction delay of any duration for any 1:1 firing pattern. The coupling can be strong as long as the effect of one input dissipates before the next input is received. We show the form that the generic synchronous and anti-phase solutions take in a system of two identical, identically pulse-coupled oscillators with identical delays. The stability criterion has a simple form that depends only on the slopes of the PRCs at the phases at which inputs are received and on the number of cycles required to complete the delayed feedback loop. The number of cycles required to complete the delayed feedback loop depends upon both the value of the delay and the firing pattern. We successfully tested the predictions of our methods on networks of model neurons. The criteria can easily be extended to include the effect of an input on the cycle after the one in which it is received.     

Circuits constructed from identified Aplysia neurons exhibit multiple patterns of persistent activity.

We have used identified neurons from the abdominal ganglion of the mollusc Aplysia to construct and analyze two circuits in vitro. Each of these circuits was capable of producing two patterns of persistent activity; that is, they had bistable output states. The output could be switched between the stable states by a brief, external input. One circuit consisted of cocultured L10 and left upper quadrant (LUQ) neurons that formed reciprocal, inhibitory connections. In one stable state L10 was active and the LUQ was quiescent, whereas in the other stable state L10 was quiescent and the LUQ was active. A second circuit consisted of co-cultured L7 and L12 neurons that formed reciprocal, excitatory connections. In this circuit, both cells were quiescent in one stable state and both cells fired continuously in the other state. Bistable output in both circuits resulted from the nonlinear firing characteristics of each neuron and the feedback between the two neurons. We explored how the stability of the neuronal output could be controlled by the background currents injected into each neuron. We observed a relatively well-defined range of currents for which bistability occurred, consistent with the values expected from the measured strengths of the connections and a simple model. Outside of the range, the output was stable in only a single state. These results suggest how stable patterns of output are produced by some in vivo circuits and how command neurons from higher neural centers may control the activity of these circuits. The criteria that guided us in forming our circuits in culture were derived from theoretical studies on the properties of certain neuronal network models (e.g., Hopfield, J. J. 1984. Proc. Natl. Acad. Sci. USA. 81:3088-3092). Our results show that circuits consisting of only two co-cultured neurons can exhibit bistable output states of the form hypothesized to occur in populations of neurons.     

The cerebellum and initiation of movement: the stretch reflex

Studies of the stretch reflex in decerebrate cats indicate a phase advance of peak sinusoidal tension in steady-state cycles between 0.1 and 10 Hz. This phase advance is reduced in acute and chronic cerebellectomy, as shown in previous investigations. Also, the augmentation of muscle peak tension in initial sinusoidal stretch cycles at 0.5-5 Hz has been found to be reduced during the time of reflex and motor instability in the several months following cerebellar ablation. This report shows the increased amplitude and phase lead of integrated electromyographic activity in initiating sinusoidal stretch cycles in the decerebrate cat. These reflex aspects are demonstrated in relation to the discharge of neurons in the dorsal spinocerebellar tract and of cerebellar cortical Purkinje cells in initial sinusoidal cycles. The intensity and phase advance of the discharge in dorsal spinocerebellar tract neurons is altered little, but these features are usually increased in Purkinje cells during initial stretches compared to continuous cycling. In terms of overall motor control, these findings are compatible with concepts of movement control, modulated by the cerebellum, in which the discharge of antagonist motor neurons is regulated in concert with that of agonist muscles upon initiation and termination of movement.     

Long-Term Stability of Visual Pattern Selective Responses of Monkey Temporal Lobe Neurons

Many neurons in primate inferotemporal (IT) cortex respond selectively to complex, often meaningful, stimuli such as faces and objects. An important unanswered question is whether such response selectivity, which is thought to arise from experience-dependent plasticity, is maintained from day to day, or whether the roles of individual cells are continually reassigned based on the diet of natural vision. We addressed this question using microwire electrodes that were chronically implanted in the temporal lobe of two monkeys, often allowing us to monitor activity of individual neurons across days. We found that neurons maintained their selectivity in both response magnitude and patterns of spike timing across a large set of visual images throughout periods of stable signal isolation from the same cell that sometimes exceeded two weeks. These results indicate that stimulus-selectivity of responses in IT is stable across days and weeks of visual experience.     

Performance of a model for a local neuron population

A model of a local neuron population is considered that contains three subsets of neurons, one main excitatory subset, an auxiliary excitatory subset and an inhibitory subset. They are connected in one positive and one negative feedback loop, each containing linear dynamic and nonlinear static elements. The network also allows for a positive linear feedback loop. The behaviour of this network is studied for sinusoidal and white noise inputs. First steady state conditions are investigated and with this as starting point the linearized network is defined and conditions for stability is discovered. With white noise as input the stable network produces rhythmic activity whose spectral properties are investigated for various input levels. With a mean input of a certain level the network becomes unstable and the characteristics of these limit cycles are investigated in terms of occurence and amplitude. An electronic model has been built to study more closely the waveforms under both stable and unstable conditions. It is shown to produce signals that resemble EEG background activity and certain types of paroxysmal activity, in particular spikes.     

Synchrony of bone marrow proliferation and maturation as the origin of cyclic haemopoiesis

Abstract. Cyclic haemopoiesis in Grey Collie dogs is characterized by stable oscillations in all haemopoietic lineages. It is proposed that in these animals, in contrast to normal animals, the maturation process of haemopoietic (in particular granuloid) cells from the primitive progenitors to the functional cells is characterized by an abnormally strong synchrony. It is conjectured that the marrow maturation time has a very small variance compared with non-cyclic normal dogs. With a mathematical model of haemopoiesis it is shown that small fluctuations are amplified via regular feedback processes such that stable granuloid oscillations are established. Erythroid oscillations are induced indirectly by granuloid feedback to the stem cell pool. The model calculations further show that the synchrony hypothesis of bone marrow maturation can quantitatively explain the following experimental results: (1) the maintenance of stable cycles of granuloid and erythroid bone marrow and blood cells with a period of approximately 14 d; (2) the disappearance of granuloid and erythroid cycles during the administration of the colony stimulating factor rhG-CSF; (3) the reappearance of oscillations when the administration of CSF is discontinued; (4) the cessation of cycles during endotoxin application; and (5) the persistence of cycles during erythroid manipulations (bleeding anaemia, hypoxia, hypertransfusion). We therefore conclude that cyclic haemopoiesis is not caused by a defect in the regulatory control system but by an unusual maturation process.     

Synchrony of bone marrow proliferation and maturation as the origin of cyclic haemopoiesis.

Cyclic haemopoiesis in Grey Collie dogs is characterized by stable oscillations in all haemopoietic lineages. It is proposed that in these animals, in contrast to normal animals, the maturation process of haemopoietic (in particular granuloid) cells from the primitive progenitors to the functional cells is characterized by an abnormally strong synchrony. It is conjectured that the marrow maturation time has a very small variance compared with non-cyclic normal dogs. With a mathematical model of haemopoiesis it is shown that small fluctuations are amplified via regular feedback processes such that stable granuloid oscillations are established. Erythroid oscillations are induced indirectly by granuloid feedback to the stem cell pool. The model calculations further show that the synchrony hypothesis of bone marrow maturation can quantitatively explain the following experimental results: (1) the maintenance of stable cycles of granuloid and erythroid bone marrow and blood cells with a period of approximately 14 d; (2) the disappearance of granuloid and erythroid cycles during the administration of the colony stimulating factor rhG-CSF; (3) the reappearance of oscillations when the administration of CSF is discontinued; (4) the cessation of cycles during endotoxin application; and (5) the persistence of cycles during erythroid manipulations (bleeding anaemia, hypoxia, hypertransfusion). We therefore conclude that cyclic haemopoiesis is not caused by a defect in the regulatory control system but by an unusual maturation process.     

Cycling in simple genetic systems

We describe the — unexpected — occurrence of stable limit cycles in the two locus, two allele model. No frequency dependence is involved. The cycles are due to the interaction between recombination and natural selection.This work received support from the National Science Foundation and the Research Foundation of the City University of New York     

Incorporating spatial variation in density enhances the stability of simple population dynamics models

Simple discrete time models of population growth admit a wide variety of dynamic behaviors, including population cycles and chaos. Yet studies of natural and laboratory populations typically reveal their dynamics to be relatively stable. Many explanations for the apparent rarity of unstable or chaotic behavior in real populations have been developed, including the possible stabilizing roles of migration, refugia, abrupt density-dependence, and genetic variation in sensitivity to density. We develop a theoretical framework for incorporating random spatial variation in density into simple models of population growth, and apply this approach to two commonly used models in ecology: the Ricker and Hassell maps. We show that the incorporation of spatial density variation into both these models has a strong stabilizing influence on their dynamic behavior, and leads to their exhibiting stable point equilibria or stable limit cycles over a relatively much larger range of parameter values. We suggest that one reason why chaotic population dynamics are less common than the simple models indicate is, these models typically neglect the potentially stabilizing role of spatial variation in density.     

Delayed feedback and multiple attractors in a host–parasitoid system

 Continuous-time, age structured, host–parasitoid models exhibit three types of cyclic dynamics: Lotka–Volterra-like consumer-resource cycles, discrete generation cycles, and “delayed feedback cycles” that occur if the gain to the parasitoid population (defined by the number of new female parasitoid offspring produced per host attacked) increases with the age of the host attacked. The delayed feedback comes about in the following way: an increase in the instantaneous density of searching female parasitoids increases the mortality rate on younger hosts, which reduces the density of future older and more productive hosts, and hence reduces the future per head recruitment rate of searching female parasitoids. Delayed feedback cycles have previously been found in studies that assume a step-function for the gain function. Here, we formulate a general host–parasitoid model with an arbitrary gain function, and show that stable, delayed feedback cycles are a general phenomenon, occurring with a wide range of gain functions, and strongest when the gain is an accelerating function of host age. We show by examples that locally stable, delayed feedback cycles commonly occur with parameter values that also yield a single, locally stable equilibrium, and hence their occurrence depends on initial conditions. A simplified model reveals that the mechanism responsible for the delayed feedback cycles in our host–parasitoid models is similar to that producing cycles and initial-condition-dependent dynamics in a single species model with age-dependent cannibalism. Received: 24 October 1997 / Revised version: 13 June 1998