Biol. Cybern. 65, 99-105 (1991) Biological Cybernetics 9 Springer-Verlag 1991 Linear systems analysis of the relationship between firing of deep cerebellar neurons and the classically conditioned nictitating membrane response in rabbits N. E. Berthier, A. G. Barto, and J. W. Moore Department of Psychology and Department of Computer and Information Science, University of Massachusetts, Amherst, MA 01003, USA Received January 7, 1991/Accepted April 9, 1991 Abstract. The correlation of the activity of neurons in the interposed and dentate nuclei of the cerebellum with conditioned movements of the nictitating membrane was investigated using linear systems analysis. The activity of single deep cerebellar nuclear cells was assumed to be the input to a linear system that produced nictitating mem- brane movement. Data were initially analyzed with a causal model to assess the degree to which past neural activity predicted the conditioned response. 55 of 165 cells had correlation coefficients of 0.50 or greater be- tween the model's moment-to-moment output and the actual output, with two interpositus cells having correla- tion coefficients of greater than 0.90. Double-sided im- pulse responses indicated that afference from the face and efference copy probably affect deep cerebellar neural activity. Nonlinearities were also found in the relation- ship between neuronal activity and conditioned move- ment. It was concluded that cerebellar deep nuclear firing is highly correlated with future nictitating mem- brane movements but that the firing-movement relation- ship contains noncausal and nonlinear components. 1 Introduction Recent investigations suggest that the cerebellum gener- ates the motor command for the nictitating membrane (NM) conditioned response in the rabbit (see Strata 1989). Recording experiments show that many cerebel- lar cortical and deep nuclear neurons are active during conditioning (Berthier and Moore 1986, 1990). Some neurons respond to the conditioned stimulus (CS), others to the unconditioned stimulus (US), and still others during conditioned responses (CRs). Some corti- cal and deep nuclear cells respond to more than one of these events during conditioning trials. If the cerebellum generates the motor command for the CR, one would expect cerebellar deep nuclear firing to be time-locked to CR-onset. Berthier and Moore (1990) investigated this possibility using variance ratios (VRs) to quantify the variability of firing-movement delays1. The VRs indicated that the firing of some cerebellar deep nuclear neurons is time-locked to CR- onset. However, VRs do not assess whether neural firing is related to the time-course or topography of the CR, a relationship that would be expected if the cere- bellum generates the CR. A method that does assess firing-movement correlations is linear time-series regres- sion or linear system identification (Houk et al. 1987; Marmarelis and Marmarelis 1978). We use this method in the present paper with two different models of the firing-movement relationship. One model only allows for neural activity to influence future positions of the NM. A good-fit of this simple model to the data would suggest that deep nuclear cells are involved in generat- ing the CR. Even if deep nuclear activity predicts future NM movement, the simple model may not fully capture the relationship of neural firing to NM movement. This is because the cerebellum receives proprioceptive input from the face and orbit (Bach-y-Rita and Ito 1966; Baker et al. 1972; Fuchs and Kornhuber 1969; Maekawa and Kimura 1980), and efference copy from extracerebellar sources (Brooks 1986; Phillips and Porter 1977). These cerebellar inputs introduce the pos- sibility that neural firing is correlated with past NM movement (i.e. firing and NM movement are non- causally correlated). In order to evaluate this possibility we extended this simple model to allow for neural firing to be correlated with previous NM movement. A theo- retical discussion of the basis of noncausal correlations in a physiological system can be found in Hunter and Kearney (1983) and Houk et al. (1987). To summarize, the present study has two aims. The first is to investigate the degree to which the firing of deep cerebellar neurons predicts the time-course of 1 VRs are the variance of the differences in time between changes in neural firing and the onsets of the CR divided by the variance of the times of change in firing with respect to the onsets of the CS (Commenges and Seal 1986)