THE SPIKE TRAINS OF INHIBITED PACEMAKER NEURONS SEEN THROUGH THE MAGNIFYING GLASS OF NONLINEAR ANALYSES

This communication describes the new information that may be obtained by applying nonlinear analytical techniques to neurobiological time-se ries. Specifically, we consider the sequence of interspike intervals T -i (the ''timing'') of trains recorded from synaptically inhibited cra yfish pacemaker neurons. As reported earlier, different postsynaptic s pike train forms (sets of timings with shared properties) are generate d by varying the average rate and/or pattern (implying interval disper sions and sequences) of presynaptic spike trains. When the presynaptic train is Poisson (independent exponentially distributed intervals), t he form is ''Poisson-driven'' (unperturbed and lengthened intervals su cceed each other irregularly). When presynaptic trains are pacemaker ( intervals practically equal), forms are either ''p:q locked'' (interva ls repeat periodically), ''intermittent'' (mostly almost locked but di srupted irregularly), ''phase walk throughs'' (intermittencies with br iefer regular portions), or ''messy'' (difficult to predict or describ e succinctly). Messy trains are either ''erratic'' (some intervals nat ural and others lengthened irregularly) or ''stammerings'' (intervals are integral multiples of presynaptic intervals). The individual spike train forms were analysed using attractor reconstruction methods base d on the lagged coordinates provided by successive intervals from the time-series T-i. Numerous models were evaluated in terms of their pred ictive performance by a trial-and-error procedure: the most successful model was taken as best reflecting the true nature of the system's at tractor. Each form was characterized in terms of its dimensionality, n onlinearity and predictability. (1) The dimensionality of the underlyi ng dynamical attractor was estimated by the minimum number of variable s (coordinates T-i) required to model acceptably the system's dynamics , i.e. by the system's degrees of freedom. Each model tested was based on a different number of T-i; the smallest number whose predictions w ere judged successful provided the best integer approximation of the a ttractor's true dimension (not necessarily an integer). Dimensionaliti es from three to five provided acceptable fits. (2) The degree of nonl inearity was estimated by: (i) comparing the correlations between expe rimental results and data from linear and nonlinear models, and (ii) t uning model nonlinearity via a distance-weighting function and identif ying the either local or global neighborhood size. Lockings were compa tible with linear models and stammerings were marginal; nonlinear mode ls were best for Poisson-driven, intermittent and erratic forms. (3) F inally, prediction accuracy was plotted against increasingly long sequ ences of intervals forecast: the accuracies for Poisson-driven, locked and stammering forms were invariant, revealing irregularities due to uncorrelated noise, but those of intermittent and messy erratic forms decayed rapidly, indicating an underlying deterministic process. The e xcellent reconstructions possible for messy erratic and for some inter mittent forms are especially significant because of their relatively l ow dimensionality (around 4), high degree of nonlinearity and predicti on decay with time. This is characteristic of chaotic systems, and pro vides evidence that nonlinear couplings between relatively few variabl es are the major source of the apparent complexity seen in these cases . This demonstration of different dimensions. degrees of nonlinearity and predictabilities provides rigorous support for the categorization of different synaptically driven discharge forms proposed earlier on t he basis of more heuristic criteria. This has significant implications . (1) It demonstrates that heterogeneous postsynaptic forms can indeed be induced by manipulating a few presynaptic variables. (2) Each pres ynaptic timing induces a form with characteristic dimensionality, thus breaking up the preparation into subsystems such that the physical va riables in each operate as one formal parameter or degree of freedom. A system's partitions differ because of component subsystems and/or dy namics: the set of all partitions is probably large and continuous. Dr iver-induced partitions have general theoretical interest, and provide guidelines for identifying the responsible physical variables. (3) Be cause forms tolerate changing conditions and are encountered widely (e .g., along transients), it is hypothesized that they are elementary bu ilding blocks for many synaptic codings. Codings are linear if postsyn aptic forms have the same spectral components as the presynaptic pacem aker, or nonlinear if novel components arise as with, respectively, 1: 1 locked or erratic trains. This is relevant to network operations whe re regularity and irregularity are often vital. (4) Rigorously identif ying spike train forms in experimental data from living preparations a llowed matchings with available theoretical computations and considera tions. Relevant models are based either on iterations of maps derived from rhythm resettings by isolated arrivals or on Bonhoeffer-van der P ol formulations: such models generate, respectively, only periodic loc king and phase walk throughs, or all forms. This precise and broad con ceptual context explains and predicts outcomes, recognizes data/theory discrepancies, and identifies their reasons (e.g., after-effects, noi se). (5) Accordingly, forms pertain to universal behavior categories c alled ''noisy'', ''periodic'', ''intermittent'', ''quasiperiodic'' or ''chaotic'' whose available theories provide valuable contexts For gen uinely physiological issues. Thus, experimental design and thinking be nefit from significant insights about the dynamics of pacemaker-driven pacemakers, the simplest of all synaptic codings. (C) 1998 IBRO. Publ ished by Elsevier Science Ltd.