## The spike trains of inhibited pacemaker neurons seen through the magnifying glass of nonlinear analyses

### Segundo, J.P. ; Sugihara, George ; Dixon, Paul ; Stiber, M. ; Bersier, Louis-Félix

### In: Neuroscience, 1998, vol. 87, no. 4, p. 741-766

This communication describes the new information that may be obtained by applying nonlinear analytical techniques to neurobiological time-series. Specifically, we consider the sequence of interspike intervals Ti (the «timing») of trains recorded from synaptically inhibited crayfish pacemaker neurons. As reported earlier, different postsynaptic spike train forms (sets of... Plus

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- This communication describes the new information that may be obtained by applying nonlinear analytical techniques to neurobiological time-series. Specifically, we consider the sequence of interspike intervals
*T*(the «timing») of trains recorded from synaptically inhibited crayfish pacemaker neurons. As reported earlier, different postsynaptic spike train forms (sets of timings with shared properties) are generated by varying the average rate and/or pattern (implying interval dispersions and sequences) of presynaptic spike trains. When the presynaptic train is Poisson (independent exponentially distributed intervals), the form is «Poisson-driven» (unperturbed and lengthened intervals succeed each other irregularly). When presynaptic trains are pacemaker (intervals practically equal), forms are either «_{i}*p:q*locked» (intervals repeat periodically), «intermittent» (mostly almost locked but disrupted irregularly), «phase walk throughs» (intermittencies with briefer regular portions), or «messy» (difficult to predict or describe succinctly). Messy trains are either «erratic» (some intervals natural and others lengthened irregularly) or «stammerings» (intervals are integral multiples of presynaptic intervals).

The individual spike train forms were analysed using attractor reconstruction methods based on the lagged coordinates provided by successive intervals from the time-series*T*. Numerous models were evaluated in terms of their predictive performance by a trial-and-error procedure: the most successful model was taken as best reflecting the true nature of the system's attractor. Each form was characterized in terms of its dimensionality, nonlinearity and predictability._{i}

(1) The dimensionality of the underlying dynamical attractor was estimated by the minimum number of variables (coordinates*T*) 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_{i}*T*; the smallest number whose predictions were judged successful provided the best integer approximation of the attractor's true dimension (not necessarily an integer). Dimensionalities from three to five provided acceptable fits._{i}

(2) The degree of nonlinearity was estimated by: (i) comparing the correlations between experimental results and data from linear and nonlinear models, and (ii) tuning model nonlinearity via a distance-weighting function and identifying the either local or global neighborhood size. Lockings were compatible with linear models and stammerings were marginal; nonlinear models were best for Poisson-driven, intermittent and erratic forms.

(3) Finally, prediction accuracy was plotted against increasingly long sequences 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 excellent reconstructions possible for messy erratic and for some intermittent forms are especially significant because of their relatively low dimensionality (around 4), high degree of nonlinearity and prediction decay with time. This is characteristic of chaotic systems, and provides evidence that nonlinear couplings between relatively few variables 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 the 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 presynaptic timing induces a form with characteristic dimensionality, thus breaking up the preparation into subsystems such that the physical variables in each operate as one formal parameter or degree of freedom. A system's partitions differ because of component subsystems and/or dynamics: the set of all partitions is probably large and continuous. Driver-induced partitions have general theoretical interest, and provide guidelines for identifying the responsible physical variables. (3) Because forms tolerate changing conditions and are encountered widely (e.g., along transients), it is hypothesized that they are elementary building blocks for many synaptic codings. Codings are linear if postsynaptic forms have the same spectral components as the presynaptic pacemaker, or nonlinear if novel components arise as with, respectively, 1:1 locked or erratic trains. This is relevant to network operations where regularity and irregularity are often vital. (4) Rigorously identifying spike train forms in experimental data from living preparations allowed matchings with available theoretical computations and considerations. Relevant models are based either on iterations of maps derived from rhythm resettings by isolated arrivals or on Bonhoeffer-van der Pol formulations: such models generate, respectively, only periodic locking and phase walk throughs, or all forms. This precise and broad conceptual context explains and predicts outcomes, recognizes data/theory discrepancies, and identifies their reasons (e.g., after-effects, noise). (5) Accordingly, forms pertain to universal behavior categories called ‘‘noisy’’, ‘‘periodic’’, ‘‘intermittent’’, ‘‘quasiperiodic’ ’or ‘‘chaotic’’ whose available theories provide valuable contexts for genuinely physiological issues. Thus, experimental design and thinking benefit from significant insights about the dynamics of pacemaker-driven pacemakers,the simples to fall synaptic codings.