Friedhelm R. Drepper

Chance and chaos in population biology—Models of recurrent epidemics and food chain dynamics

Abstract In spite of their complex behaviour the dynamics of some ecological systems may be explained by deterministic laws of motion. This is clearly theoretically possible, because deductive approaches to the dynamics of ecological systems lead to nonlinear models and even very simple model ecosystems generate typically a rich spectrum of dynamics, ranging from coexisting periodic regimes to chaotic behaviour. The present study analyses the complex interplay between deterministic nonlinear dynamics and demographic stochasticity using two examples. Firstly, the search for chaos in ecology has drawn much attention to the analysis of recurrent outbreaks of childhood epidemics—in particular of measles infections—in large population centres. The second example is a compartmental model of a three-species food chain, where sudden population disappearances can be observed due to transient chaos. Our analysis lends support to the plausibility of chaos in population dynamics. However, for decreasing population size (increasing fluctuations) the distinction between chaos and stochasticity becomes more and more problematic. The interplay between deterministic dynamics and stochastic fluctuations, e.g. due to the integer structure of populations or a noisy environment, suggests an analysis combining methods from the theory of nonlinear dynamical systems as well as stochastic processes.

One-dimensional, nonlinear determinism characterizes heart rate pattern during paced respiration

This study focuses on the dynamic pattern of heart rate variability in the frequency range of respiration, the so-called respiratory sinus arrhythmia. Forty experimental time series of heart rate data from four healthy adult volunteers undergoing a paced respiration protocol were used as an empirical basis. For pacing-cycle lengths >8 s, the heartbeat intervals are shown to obey a rule that can be expressed by a one-dimensional circle map (next-angle map). Circle maps are introduced as a new type of model for time series analyses to characterize the nonlinear dynamic pattern underlying the respiratory sinus arrhythmia during voluntary paced respiration. Although these maps are not chaotic, the dynamic pattern shows typical imprints of nonlinearity. By starting from a piecewise linear model, which describes the different circle maps obtained from the empirical time series for various pacing frequencies, time invariant measures can be introduced that characterize the dynamic pattern of heart rate variability during voluntary slow-paced respiration.

Nonlinear time series analysis of empirical population dynamics

The classical approach towards time series analysis of fluctuating phenomena is based on linear stochastic processes in the basin of attraction of globally stable equilibria (ARMA models). However, there are deductive reasons derived from nonequilibrium statistical mechanics that many self-generated population fluctuations should be interpreted as stochastic processes influenced by nonequilibrium attractors of nonlinear dynamical systems. The incidence pattern of measles epidemics in New York City is used to demonstrate that highly nonlinear autoregressive models can serve as a new semi-phenomenological level of description for complex self-generated fluctuations in biological or ecological systems. The nature of the unpredictability of the incidence pattern of measles is characterized by a subtle interaction between a chaotic nonlinear determinism and stochastic fluctuations. The sensitive dependence on initial conditions, characteristic for chaos, is not limited to chaotic attractors but can also occur on so-called chaotic transients. The dynamics of recurrent outbreaks of measles in New York City turns out to be close to a so-called boundary crisis which converts a stable chaotic attractor to metastable chaotic transients leading finally to a period-3 attractor. However, the demographic noise destabilizes the periodic orbit as well, and creates a situation of intermittent jumps between episodic periodicities and longer chaotic transients. A simple autoregressive model is used to achieve a plausible geometric understanding of the noise-induced intermittency switching between episodic periodicity and transient chaos.

Qualitative Analysis of Unpredictability: A Case Study from Childhood Epidemics

The unpredictability of the recurrent outbreaks of childhood epidemics has been a matter of scientific dispute. At first sight the deterministic SEIR model, which divides the host population into four classes (Susceptible, Exposed, Infectious, Recovered), fails to explain the unpredictability, because realistic parameter values lead to periodic attractors. We show that these periodic attractors coexist with chaotic transients. The detailed geometrical analysis of this phenomenon suggests that chaotic transients have been underestimated in their importance for the dynamics on observable time scales. A second problem of the SEIR model is the high extinction probability of epidemics in finite populations. We argue that the immigration of infectives from outside is an essential parameter in this context. Immigration and the process of infection itself are sources of demographic fluctuations, which undergo subtle interaction with chaotic transients. The stochastic simulation of the SEIR model shows that the chaotic transients are permanently revisited. The demographic noise integrates chaotic transients and intermittent periodic episodes.

Non-stationary feature extraction for automatic speech recognition

In current speech recognition systems mainly Short-Time Fourier Transform based features like MFCC are applied. Dropping the short-time stationarity assumption of the voiced speech, this paper introduces the non-stationary signal analysis into the ASR framework. We present new acoustic features extracted by a pitch-adaptive Gammatone filter bank. The noise robustness was proved on AURORA 2 and 4 tasks, where the proposed features outperform the standard MFCC. Furthermore, successful combination experiments via ROVER indicate the differences between the new features and MFCC.

A two-level drive – response model of non-stationary speech signals

The transmission protocol of voiced speech is hypothesized to be based on a funda mental drive process, which synchronizes the vocal tract excitation on the trans mitter side and evokes the pitch perception on the receiver side. A band limited fundamental drive is extrac ted from a voice specific subband decom position of the speech signal. When the near periodic drive is used as fun damental drive of a two-level drive-response model, a more or less aperiodic voiced excitation can be recon struc ted as a more or less aperiodic trajectory on a low dimensional continuous syn chro nization manifold (surface) described by speaker and phoneme specific coupling functions. In the case of vowels and nasals the excitation can be described by a univariate coupling function, which depends on the momentary phase of the funda mental drive. In the case of other voiced consonants the coupling function may as well depend on a delayed funda mental phase with a phoneme speci fic time delay. The delay may exceed the length of the analysis window. The resulting long range correlation cannot be analysed or synthesized by models assuming stationary excitation.

Voiced speech as response of a self-consistent fundamental drive

Voiced segments of speech are assumed to be composed of non-stationary acoustic objects which can be described as stationary response of a non-stationary fundamental drive (FD) process and which are furthermore suited to reconstruct the hidden FD by using a voice adapted (self-consistent) part-tone decomposition of the speech signal. The universality and robustness of human pitch perception encourage the reconstruction of a band-limited FD in the frequency range of the pitch. The self-consistent decomposition of voiced continuants generates several part-tones which can piecewise be confirmed to be topologically equivalent to corresponding acoustic modes of the excitation on the transmitter side. As topologically equivalent image of a glottal master oscillator, the self-consistent FD is suited to serve as low frequency part of the basic time-scale separation of auditive perception and to describe the broadband voiced excitation as entrained (synchronized) and/or modulated primary response. Being guided by the acoustic correlates of pitch and loudness perception, the time-scale separation avoids the conventional assumption of stationary excitation and represents the basic decoding step of an advanced precision transmission protocol of self-consistent (voiced) acoustic objects. The present study is focussed on the adaptation of the trajectories (contours) of the centre filter frequency of the part-tones to the chirp of the glottal master oscillator.

Phase difference of filter-stable part-tones as acoustic feature

A part-tone decomposition of voiced sections of speech is introduced, which is adapted with high accuracy to the frequency of the glottal oscillator of the speaker. The iterative replacement of the center filter frequency contours (chosen locally as linear chirp) of the non-stationary bandpass filters converges extremely fast and leads to the extraction of filter-stable part-tones with uncorrupted phases. In contrast to phases of frequency decomposition with a priori defined, constant filter frequencies, the phase differences of filter-stable part-tones promise to become a useful supplement of the amplitude based acoustic features used for conventional automatic speech recognition. The derived phase features are tested in vowel classification experiments based on the phonetically rich TIMIT database.

Non-stationary self-consistent acoustic objects as atoms of voiced speech

To account for the strong non-stationarity of voiced speech and its nonlinear aero-acoustic origin, the classical source-filter model is extended to a cascaded drive-response model with a conventional linear secondary response, a synchronized and/or synchronously modulated primary response and a non-stationary fundamental drive which plays the role of the long time-scale part of the basic time-scale separation of acoustic perception. The transmission protocol of voiced speech is assumed to be based on non-stationary acoustic objects which can be synthesized as the described secondary response and which are analysed by introducing a self-consistent (filter stable) part-tone decomposition, suited to reconstruct the hidden fundamental drive and to confirm its topological equivalence to a glottal master oscillator. The filter-stable part-tone decomposition opens the option of a phase modulation transmission protocol of voiced speech. Aiming at communication channel invariant acoustic features of voiced speech, the phase modulation cues are expected to be particularly suited to extend and/or replace the classical feature vectors of phoneme and speaker recognition.