SCALING OF PREDICTION ERROR DOES NOT CONFIRM CHAOTIC DYNAMICS UNDERLYING IRREGULAR FIRING USING INTERSPIKE INTERVALS FROM MIDBRAIN DOPAMINE NEURONS C. C. CANAVIER, a * S. R. PERLA a AND P. D. SHEPARD b a Department of Psychology, University of New Orleans, GP2001, 2000 Lakeshore Drive, New Orleans, LA 70471, USA b Maryland Psychiatric Research Center, Baltimore, MD 21228, USA Abstract—Dopamine neurons in the substantia nigra pars compacta often fire in an irregular, single spike mode in vivo, and a similar firing pattern can be observed in vitro when small conductance calcium-activated potassium channel blockers are applied to the bath. It is not clear whether the irregular firing is due to stochastic processes or nonlinear deterministic processes. A previous study [Neuroscience 104 (2001) 829] used nonlinear forecasting methods applied to a continuous function derived from the interspike interval (ISI) data from irregularly firing dopamine neurons to show that the predictability scaled exponentially with forecast horizon and was consistent with nonlinear deterministic chaos. How- ever, we show here that the observed exponential scaling is also consistent with a stochastic process, because it did not differ significantly from that of shuffled surrogate data. On the other hand, nonlinear forecasting directly from the ISI data using the package TISEAN provided some evidence for nonlinear deterministic structure in four of five records ob- tained in vitro and in two of nine records obtained in vivo. Although we cannot rule out a role for nonlinear chaotic dynamics in structuring the firing pattern, we suggest an alternate hypothesis that includes a mechanism by which the firing pattern can become more variable in the presence of a constant level of background noise. © 2004 IBRO. Published by Elsevier Ltd. All rights reserved. Key words: nonlinear forecasting, pacemaker, noise, firing pattern, substantia nigra, temporal coding. A recent experimental study on the firing patterns of mid- brain dopamine cells in freely-moving, unanesthetized rats (Hyland et al., 2002) revealed a continuum of firing modes with respect to both the regularity of firing and the bursti- ness of the firing pattern. Some neurons fired in an irreg- ular fashion while still others fired a high proportion of their spikes in bursts, most often with two to three spikes. In addition, a sizeable minority of the dopamine neurons in unanesthetized animals fired in an extremely regular, clock-like activity pattern. Tepper et al. (1995) also ob- served regular rhythmic firing in vivo, and noted transitions between firing modes following manipulation of afferent inputs. In a slice preparation, dopamine neurons generally fire in a rhythmic, pacemaker-like fashion. The slice prep- aration is lacking most afferents, including the glutamater- gic afferents often associated with burst firing in these neurons (Overton and Clark, 1997). The small conduc- tance (SK) calcium-activated potassium channel is respon- sible for the afterhyperpolarization that follows an action potential during pacemaker firing (Shepard and Bunney, 1988). Apamin blocks the SK channel, and the bath appli- cation of apamin to cells firing in a pacemaker-like fashion in vitro can convert the firing pattern to bursting or irregular firing (Gu et al., 1992; Ping and Shepard, 1996). In this study, we have focused on data from spontaneously irreg- ular firing neurons in anesthetized rats, and on irregular firing induced in the slice preparation by the application of apamin to the bath (see Fig. 1). Application of apamin greatly increases the coefficient of variation (CV) in the slice preparation (Shepard and Stump, 1999; Wolfart et al., 2001). The available data consist of sets of interspike intervals (ISI), and here we tried to solve the inverse prob- lem, that is, to determine the nature of underlying dynam- ical process that generated these sequences of ISI. In particular, we seek to determine whether the under- lying dynamics are deterministic or stochastic. A determin- istic system is governed by a set of rules for how each state variable in the system will change in time based only upon the current values of each of those state variables. If the dependence of the rate of change on any variable is nonlinear, then the system is also nonlinear. A property of a deterministic system is that if you could accurately mea- sure the value of each of the variables at any point in time, you could in principle determine the value of each variable at any time in the future. In practice, nonlinear systems may have a very sensitive dependence on initial conditions such that the accuracy of the best possible prediction decreases with increasing forecast horizon, because it is impossible to simultaneously measure these quantities with infinite precision. Nonlinear systems can exhibit de- terministic chaos. The dynamics of such a system can best be visualized in a space with one dimension for each state variable. The set of values that the system can assume is restricted to a finite volume within the space, on a surface called a chaotic attractor, but the surface is so complex and convoluted that a given point is never revisited, mak- ing the period of a chaotic oscillation infinite. However, the dynamics are not without order. Nearby trajectories remain close for a finite period of time, but a hallmark of chaos is that they diverge exponentially. Thus, the accuracy of the *Corresponding author. Tel: +1-504-280-6775; fax: +1-504-280-6049. E-mail address: ccanavie@uno.edu (C. C. Canavier). Abbreviations: CV, coefficient of variation; IPSC, inhibitory postsynap- tic current; ISI, interspike interval; r, Pearson’s correlation coefficient; r s , Spearman’s rank correlation coefficient; SDF, spike density func- tion; SK, small conductance; TISEAN, TIme SEries ANalysis. Neuroscience 129 (2004) 491–502 0306-4522/04$30.00+0.00 © 2004 IBRO. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.neuroscience.2004.08.003 491