Leslie matrix models as “stroboscopic snapshots” of McKendrick PDE models

 High dimensional Leslie matrix models have long been viewed as discretizations of McKendrick PDE models. However, these two fundamental classes of models can be linked in a completely different way. For populations with periodic birth pulses, Leslie models of any dimension can be viewed as “stroboscopic snapshots” (in time) of an associated impulsive McKendrick model; that is, the solution of the discrete model matches the solution of the corresponding continuous model at every discrete time step. In application, McKendrick models of populations with birth pulses can be used to identify the state of the population between the discrete census times of the associated Leslie model. Furthermore, McKendrick models describing populations with near-synchronous birth pulses can be viewed as realistic perturbations of the associated Leslie model. Received: 7 August 1997 / Revised version: 15 January 1998     

Pulsatile signaling in intercellular communication. Periodic stimuli are more efficient than random or chaotic signals in a model based on receptor desensitization.

The efficiency of various patterns of pulsatile stimulation is determined in a model in which a receptor becomes desensitized in the presence of its stimulatory ligand. The effect of stochastic or chaotic changes in the duration and/or interval between successive pulses in a series of square-wave stimuli is investigated. Before addressing the effect of random variations in the pulsatile signal, we first extend the results of a previous analysis (Li, Y.X., and A. Goldbeter. 1989. Biophys. J. 55:125-145) by demonstrating the existence of an optimal periodic signal that maximizes target cell responsiveness whatever the magnitude of stimulation. As to the effect of stochastic or chaotic variations in the pulsatile stimulus, three kinds of random distributions are used, namely, a Gaussian and a white-noise distribution, and a chaotic time series generated by the logistic map. All these random distributions are symmetrically centered around the reference value of the duration or interval that characterizes the optimal periodic stimulus yielding maximal responsiveness in target cells. Stochastically or chaotically varying pulses are less effective than the periodic signal that corresponds to the optimal pattern of pulsatile stimulation. The response of the receptor system is most sensitive to changes in the time interval that separates successive stimuli. Similar conclusions hold when stochastic or chaotic signals are compared to a reference periodic stimulus differing from the optimal one, although the effect of random variations is then reduced. The decreased efficiency of stochastic pulses accounts for the observed superiority of periodic versus stochastic pulses of cyclic AMP (cAMP) in Dictyostelium amoebae. The results are also discussed with respect to the efficiency of periodic versus stochastic or chaotic patterns of hormone secretion.     

Extracellular transduction events under pulsed stimulation in moth olfactory sensilla

In natural conditions, pheromones released continuously by female moths are broken in discontinuous clumps and filaments. These discontinuities are perceived by flying male moths as periodic variations in the concentration of the stimulus, which have been shown to be essential for location of females. We study analytically and numerically the evolution in time of the activated pheromone-receptor (signaling) complex in response to periodic pulses of pheromone. The 13-reaction model considered takes into account the transport of pheromone molecules by pheromone binding proteins (PBP), their enzymatic deactivation in the perireceptor space and their interaction with receptors at the dendritic membrane of neurons in Antheraea polyphemus sensitive to the main pheromone component. The time-averaged and periodic properties of the temporal evolution of the signaling complex are presented, in both transient and steady states. The same time-averaged response is shown to result from many different pulse trains and to depend hyperbolically on the time-averaged pheromone concentration in air. The dependency of the amplitude of the oscillations of the signaling complex on pulse characteristics, especially frequency, suggests that the model can account for the ability of the studied type of neuron to resolve repetitive pulses up to 2 Hz, as experimentally observed. Modifications of the model for resolving pulses up to 10 Hz, as found in other neuron types sensitive to the minor pheromone components, are discussed.     

Bifurcations in the decremental propagation of a spike train in the Hodgkin-Huxley model of low excitability

Response of a nerve fiber of low excitability to periodic stimulus pulses is studied with computer simulation of the Hodgkin-Huxley model. The excitability of the Hodgkin-Huxley model is reduced by decreasing the equilibrium potential for the sodium ion and by increasing the temperature, so that the decremental propagation of spikes occurs in the refractory period. It is shown that, as the period of stimulus pulses is decreased, the propagation length of the spikes is continuously changed, and period-doubling bifurcations occur. The response of a nerve fiber of low excitability is then qualitatively different from that of a normal fiber. Received: 6 December 1996 / Accepted in revised form: 12 June 1998     

Stochastic Properties of Discrete Waves of the Limulus Photoreceptor

In the dark-adapted photoreceptor of the horseshoe crab, Limulus, transient discrete depolarizations of the cell membrane, discrete waves, occur in total darkness and their rate of occurrence is increased by illumination. The individual latencies of the discrete waves evoked by a light stimulus often cannot be resolved because the discrete waves overlap in time. The latency of the first discrete wave that follows a stimulus can be determined with reasonable accuracy. We propose a model which allows us to make an estimate of the distribution of the latencies of the individual light-evoked discrete waves, and to predict the latency distribution of the first discrete wave that follows a stimulus of arbitrary intensity-time course from the latency distribution of the first discrete wave that follows a brief flash of light. For low intensity stimuli, the predictions agree well with the observations. We define a response as the occurrence of one or more discrete waves following a stimulus. The distribution of the peak amplitudes of responses suggests that the peak amplitude of individual discrete waves sometimes has a bimodal distribution. The latencies of the two types of discrete waves, however, follow similar distributions. The area under the voltage-time curve of responses that follow equal energy long (1.25 sec) and short (10 msec) light stimuli follows similar distributions, and this suggests that discrete waves summate linearly.     

Automatic Realistic Real Time Stimulation/Recording in Weakly Electric Fish: Long Time Behavior Characterization in Freely Swimming Fish and Stimuli Discrimination

Weakly electric fish are unique model systems in neuroethology, that allow experimentalists to non-invasively, access, central nervous system generated spatio-temporal electric patterns of pulses with roles in at least 2 complex and incompletely understood abilities: electrocommunication and electrolocation. Pulse-type electric fish alter their inter pulse intervals (IPIs) according to different behavioral contexts as aggression, hiding and mating. Nevertheless, only a few behavioral studies comparing the influence of different stimuli IPIs in the fish electric response have been conducted. We developed an apparatus that allows real time automatic realistic stimulation and simultaneous recording of electric pulses in freely moving Gymnotus carapo for several days. We detected and recorded pulse timestamps independently of the fish’s position for days. A stimulus fish was mimicked by a dipole electrode that reproduced the voltage time series of real conspecific according to previously recorded timestamp sequences. We characterized fish behavior and the eletrocommunication in 2 conditions: stimulated by IPIs pre-recorded from other fish and random IPI ones. All stimuli pulses had the exact Gymontus carapo waveform. All fish presented a surprisingly long transient exploratory behavior (more than 8 h) when exposed to a new environment in the absence of electrical stimuli. Further, we also show that fish are able to discriminate between real and random stimuli distributions by changing several characteristics of their IPI distribution.     

The double effect of myocardial stimulation

The stimulus generated at the sinus node induces on the heart working cells a stronger contraction at higher frequences. The purpose of the present report is to investigate whether the two effects a) exicitation-contraction coupling and b) contraction strength are independent. Instead of the pace-maker it has been used a rectangular waves generator; the pulses are applied to isolated guinea-pig auricle or rat right ventricle. The AA. show a stronger twitch of cell when the membrane is depolarized without contraction. This phenomenon can be obtained as follows: by applying peculiar conditions of high frequency train pulses (so that they are not capable of eliciting contraction) or depolarizing the membrane with high KO concentration. The AA. believe that the stronger contraction of the muscle cell is due to the depolarization generated by the stimulus and it is not connected to the contraction. The potentiating factor is independent by cAMP and cytoplasmic Ca++. This factor is antagonized by propranolol.     

On the role of subthreshold dynamics in neuronal signaling

The role of subthreshold dynamics in neuronal signaling is examined using periodic pulse train stimulation of the Fitzhugh-Nagumo (FN) model of nerve membrane excitability and results from the squid giant axon as an experimental data base. For a broad range of stimulus conditions the first pulse in a pulse train elicited an action potential, whereas all subsequent pulses elicited subthreshold responses, both in the axon and in the FN model. These results are not well described by the Hodgkin and Huxley 1952 model. Various different patterns of subthreshold responses, including chaotic dynamics, can be observed in both systems-the FN model and the axon-depending upon stimulus conditions. For some conditions action potentials are occasionally interspersed among the subthreshold events with randomly occurring interspike intervals. The randomness is directly attributable to the underlying subthreshold chaos-deterministic chaos-rather than to a stochastic noise source. We conclude that this mechanism may contribute to multimodal interspike interval histograms which have been observed from individual neurons throughout the nervous system.     

Temporal correlations and coding of information in the visual cortex of the cat

We have observed repeated patterns in evoked spike trains recorded from the primary visual cortex of the cat. These patterns are called "triplets" and "ghost doublets". Triplets are groups of three pulses, that may or may not be adjacent to one other, the mutual intervals of which are replicated in one other group of three spikes with a precision higher than 0.15 ms. Ghost doublets are doublets of pulses whose interval replicates, with the above precision, one of the intervals of the repeated triplets and are also present in the record. In one of the 9 recorded cells, in which pulses were clearly emitted in bursts in phase with the drifting of the sinusoidal grating used as a stimulus, we could show that local temporal correlations in the form of replicating triplets and ghost doublets correspond very precisely to the temporal phase of the grating: the study of the distance between triplets, or between triplets and ghost doublets, gives a remarkably precise value of the time frequency of the grating.     

Noise-induced neural impulses

The firing pattern of neural pulses often show the following features: the shapes of individual pulses are nearly identical and frequency independent; the firing frequency can vary over a broad range; the time period between pulses shows a stochastic scatter. This behaviour cannot be understood on the basis of a deterministic non-linear dynamic process, e.g. the Bonhoeffer-van der Pol model. We demonstrate in this paper that a noise term added to the Bonhoeffer-van der Pol model can reproduce the firing patterns of neurons very well. For this purpose we have considered the Fokker-Planck equation corresponding to the stochastic Bonhoeffer-van der Pol model. This equation has been solved by a new Monte Carlo algorithm. We demonstrate that the ensuing distribution functions represent only the global characteristics of the underlying force field: lines of zero slope which attract nearby trajectories prove to be the regions of phase space where the distributions concentrate their amplitude. Since there are two such lines the distributions are bimodal representing repeated fluctuations between two lines of zero slope. Even in cases where the deterministic Bonhoeffer-van der Pol model does not show limit cycle behaviour the stochastic system produces a limit cycle. This cycle can be identified with the firing of neural pulses.     

Discrete and continuous modes of curved-line discrimination controlled by effective stimulus duration

In previous experiments two extreme modes of visual discrimination performance have been investigated by measuring small differences in pattern shape at points along a continuum of pattern shapes. These two modes, associated with discrete and continuous encoding processes, were obtained by simultaneously manipulating the number of pattern components in the display and the effective duration of the display. In this experiment, discrimination performance was measured for a fixed number of pattern components, namely three, and variable display time course. The stimuli used were curved lines drawn from a continuum with curvature parameter s. There were three stimulus time courses: (1) 2-s stimulus display, (2) 100-ms stimulus display, and (3) 100-ms stimulus display followed by a post-stimulus mask. Discrimination performance declined smoothly and monotonically with s for (1), but varied non-monotonically with s revealing a central peak for (3). Performance for (2) was intermediate between that for (1) and that for (3). A reduction in effective stimulus duration alone was thus sufficient to cause a transition from continuous to discrete modes of discrimination performance, a result which may be compatible with an explanation of variable discrimination modes based on a method of successive internal approximations of the stimulus.     

Auditory Nerve Spike Generator Modeled as a Variable Attenuator Based on a Saddle Node on Invariant Circle Bifurcation

Mammalian inner hair cells transduce the sound waves amplified by the cochlear amplifier (CA) into a graded neurotransmitter release that activates channels on auditory nerve fibers (ANF). These synaptic channels then charge its dendritic spike generator. While the outer hair cells of the CA employ positive feedback, poising on Andronov-Hopf type instabilities which make them extremely sensitive to faint sounds and make CA output strongly nonlinear, the ANF appears to be based on different principles and a different type of dynamical instability. Its spike generator “digitizes” CA output into trains of action potentials and behaves as a linear filter, rate-coding sound intensity across a wide dynamic range. Here we model the spike generator as a 3 dimensional version of a saddle node on invariant circle (SNIC) bifurcation. The generic 2d SNIC increases its spike rate as the square root of the input current above its spiking threshold. We add negative feedback in the form of a low voltage-threshold potassium conductance that slows down the generator’s rate of increase of its spike rate. A Poisson random source simulates an inner hair cell, outputting a series of noisy periodic current pulses to the model ANF whose spikes phase lock to these pulses and have a linear frequency to current relation with a wide dynamic range. Also, the spike generator compartment has a cholinergic feedback connection from the olive and experiments show that such feedback is able to alter the amount of H conductance inside the generator compartment. We show that an olive able to decrease H would be able to shift the spike generator’s dynamic range to higher sound intensities. In a quiet environment by increasing H the olive would be able to make spike trains similar to those caused by synaptic input.     

Frequency encoding of pulsatile signals of cAMP based on receptor desensitization in Dictyostelium cells

Dictyostelium discoideum amoebae represent a prototype for the study of periodic signaling in intercellular communication. These cells synthesize cAMP in response to cAMP pulses. Cell responsiveness in Dictyostelium can be characterized by the capability to generate a large number of significant responses to cAMP signals in a given amount of time. The existence of a frequency of pulsatile cAMP signals yielding maximum responsiveness is demonstrated by analysis of a realistic model for cAMP synthesis, based on receptor desensitization. The optimal frequency of stimulation closely depends on the kinetics of receptor desensitization and resensitization in target cells. Synthesis of cAMP is determined both in conditions where cells are not excitable and in conditions where they relay suprathreshold pulses of cAMP. Moreover, the effect of the stimulus waveform is investigated, and several measures of cell responsiveness are compared. The results provide an explanation for the effectiveness of cAMP pulses delivered at 5 min intervals, and for the failure of pulses delivered at 2 min intervals, in inducing slime mold development. Besides applying to intercellular communication in Dictyostelium, the present analysis bears on patterns of pulsatile signaling observed for hormones and growth factors. In all these cases, it appears that pulsatile signals can be encoded in terms of their frequency on the basis of desensitization in target cells.     

Modelling the isometric force response to multiple pulse stimuli in locust skeletal muscle

An improved model of locust skeletal muscle will inform on the general behaviour of invertebrate and mammalian muscle with the eventual aim of improving biomedical models of human muscles, embracing prosthetic construction and muscle therapy. In this article, the isometric response of the locust hind leg extensor muscle to input pulse trains is investigated. Experimental data was collected by stimulating the muscle directly and measuring the force at the tibia. The responses to constant frequency stimulus trains of various frequencies and number of pulses were decomposed into the response to each individual stimulus. Each individual pulse response was then fitted to a model, it being assumed that the response to each pulse could be approximated as an impulse response and was linear, no assumption were made about the model order. When the interpulse frequency (IPF) was low and the number of pulses in the train small, a second-order model provided a good fit to each pulse. For moderate IPF or for long pulse trains a linear third-order model provided a better fit to the response to each pulse. The fit using a second-order model deteriorated with increasing IPF. When the input comprised higher IPFs with a large number of pulses the assumptions that the response was linear could not be confirmed. A generalised model is also presented. This model is second-order, and contains two nonlinear terms. The model is able to capture the force response to a range of inputs. This includes cases where the input comprised of higher frequency pulse trains and the assumption of quasi-linear behaviour could not be confirmed.     

Frequency specificity in intercellular communication. Influence of patterns of periodic signaling on target cell responsiveness.

Cells often communicate by means of periodic signals, as exemplified by a large number of hormones and by the aggregation of Dictyostelium discoideum amebas in response to periodic pulses of cyclic AMP. Periodic signaling allows bypassing the phenomenon of desensitization brought about by constant stimuli. To gain further insight into the efficiency of pulsatile signaling, we analyze the effect of periodic stimulation on the dynamic behavior of a receptor system capable of desensitization toward its ligand. We first show that the receptor system adapts to square-wave stimuli, i.e., the response eventually reaches a steady, periodic pattern after a transient phase. By analyzing the dependence of the response on the characteristics of the square-wave stimulation, we show that there exist a waveform and a period of that signal that result in maximum responsiveness of the target system. Similar results are obtained when the signal takes the more realistic form of a periodically repeated stimulation followed by exponential decay of the ligand. The results are discussed with respect to the role of pulsatile secretion of gonadotropin-releasing hormone (GnRH) by the hypothalamus and of periodic signaling by cyclic AMP pulses in Dictyostelium. The analysis accounts for the existence, in both cases, of an optimal frequency and waveform of the periodic stimulus that correspond to maximum target cell responsiveness.     

On phase locking of pulse encoders

Several models have been proposed to underststand the patterns of nerve impulses produced by periodic stimuli. This paper shows that for a very large class of such models there exists a pattern of phases that repeats periodically after a finite number of pulses; the actual pulses produced by the model depend on its initial condition, but in all cases they either follow such a pattern or approach it asymptotically. This paper is dedicated to the memory of M. G. F. Fuortes, neurophysiologist, who honored me with his friendship     

Analysis of cyclic fluctuations in larch bud moth populations by means of discrete-time dynamic models

Analysed are the data of larch bud moth (Zeiraphera diniana Gn.) fluctuations in Swiss Alps. The analysis applies simplest mathematical models of isolated population dynamics (in particular, Kostitzin model, Skellam model, the discrete logistic model, and some other ones), which include the minimal number of unknown parameters. The parameters have been estimated, for all the models in hand, by the least-squares method, to fit certain data from the Global Population Dynamics Database (N 1407 and N 6195), the sequences of the data deviations from the model trajectories being treated as well. The best approximations are shown to be achieved with Moran-Ricker model and the discrete logistic model. Statistical criteria (Kolmogorov-Smirnov and Shapiro-Wilk tests) reveal that the hypotheses of normal distribution of residuals must be rejected for one of the time series (N 1407); some models demonstrate serial correlations in the sequence of residuals (according to Durbin-Watson test). This leads to the conclusion that periodic fluctuations in the larch bud moth population (N 1407) can hardly be explained by self-regulation mechanisms alone. For another time series (N 6195), the modified discrete logistic model has appeared to be acceptable as a mode of fluctuations.     

Modeling population responses of rapidly-adapting mechanoreceptive fibers

The population response of rapidly-adapting (RA) fibers is one component of the physiological substrate of the sense of touch. Herein, we describe a computational scheme based on the population-response model by K.O. Johnson (J. Neurophysiol. 37: 48–72, 1974) which we extended by permitting the capability to include the spatial distributions of receptors in the glabrous skin linked to RA fibers. The hypothetical cases simulated were rectangular, uniformly random and proximo-distally Gaussian distributions. Each spatial organization produced qualitatively distinct population-response profiles that also varied due to stimulus parameters. The effects of stimulus amplitude, average innervation density and contactor-probe location were studied by considering various response measures: number of active fibers, summated firing rate and the average firing rate of a subset of the modeled population. The outcome of the measures were statistically compared among simulated anatomical distributions. The response is the same for rectangular and uniformly random distributions, both of which have a homogeneous innervation density. However, the Gaussian distribution produced statistically different responses when the measure was not averaged over the subset population which represented the receptive field of a higher-order neuron. These results indicate that, as well as stimulus parameters, the anatomical organization is a significant determinant of the population response. Therefore, reconstructing population activity for testing psychophysical hypotheses must presently be done with care until the organization of the receptors within the skin has been clarified.