Rat Behavior Detection And Video System Limitations

Psychopharmacology (1993) 113 : 177 186

Psychopharmacology © Springer-Verlag 1993

Quantitative assessment of the microstructure of rat behavior:

I. f(d), The extension of the scaling hypothesis* Martin P. Paulus l'z, Mark A. Geyer 2 1Laboratory of Biological Dynamics and Theoretical Medicine, University of California, San Diego (0804) La Jolla, California 92093, USA 2 Department of Psychiatry, School of Medicine, University of California, San Diego (0804) La Jolla, California 92093, USA Received October 6, 1992/Final version April 20, 1993

Abstract. Previous studies demonstrated that drug effects on the movement sequences of rats in unconditioned motor activity paradigms can be quantified by scaling measures that describe the average relationship between a variable of interest and an experimental parameter. However, rats engage in a wide variety of geometrically distinct movements that can be influenced differentially by drugs. In this investigation, the extended scaling approach is presented to capture quantitatively the relative contributions of geometrically distinct movement sequences to the overall path structure. The calculation of the spectrum of local spatial scaling exponents, f(d), is based on ensemble methods used in statistical physics. Results of the f(d) analysis confirm that the amount of motor activity is not correlated with the geometrical structure of movement sequences. Changes in the average spatial scaling exponent, d, correspond to shifting the entiref(d) function, and indicate overall changes in path structure. With the extended scaling approach, straight movement sequences are assessed independently from highly circumscribed movements. Thus, the rid) function identifies drug effects on particular ranges of movement sequences as defined by the geometrical structure of movements. More generally, the f(d) function quantifies the relationship between microscopically recorded variables, in this paradigm consecutive (X, y) locations, and the macroscopic behavioral patterns that constitute the animal's response topography. Key words: Rat - Behavior - Microstructure measures

Scaling

The assessment of unconditioned motor behavior in animals has been used as a quantitative descriptor of arousal. * Exemplary calculations of path segments with different geometrical characteristics can be obtained from the authors. Questions regarding the computational implementation of this method should be addressed to MPP preferably via email ([email protected]) or FAX (619-543-2493) Correspondence to: M.P. Paulus

Arousal refers to a state of awakeness related to the activity of the ascending reticular formation (Hull 1949; Hebb 1955; Robbins and Everitt 1990). Based on measures of the amount of motor activity in rodents, pharmacological agents can be categorized as psychomotor stimulants or depressants (Harvey 1987). However, different types of behavior can be associated with similar amounts of activity (Geyer et al. 1986). Moreover, it has been proposed that arousal is not a simple scalar state but can be differentiated by distinct patterns of activity (Robbins and Everitt 1990). Consequently, the assessment of arousal requires the quantification of different patterns of unconditioned motor activity to identify behavioral profiles that may help to determine the modes and sites of action of these substances. Unconditioned motor activity is frequently measured by counting interruptions of photobeams in a standardized testing environment (Robbins 1977; Reiter and MacPhail 1979; Geyer 1990). In some cases, traveled distances have also been used to quantify the amount of motor activity (Sanberg et al. 1987). However, when measuring the amount of motor activity, instrument-related features crucially affect the measurement. The most important of these features include the distance between photobeams, the resolution used to measure the movements in video devices, and the sampling frequency. Several investigators (Ljungberg and Ungerstedt 1977; Nickolson 1981; Geyer et al. 1986; Sanberg et al. 1987; Stahle 1987) have distinguished between different types of motor behavior by introducing a priori definitions of different movement patterns. Others (Fray et al. 198t) have used rating scales to quantify different categories of behavior. Both approaches acknowledge that the amount of motor activity provides only a partial description of the behavior, which suggests that measures of patterns of motor activity yield important additional information. Traditionally, the assessment of behavioral patterns are based on behavioral categories that are defined a priori. However, we (Paulus and Geyer 1991a) and others (Szechtman et al. 1988) have argued that discrete behavioral categories are not adequate descriptors for the micro-analysis of unconditioned motor behavior. First. it

178 is frequently difficult to compare the behavioral effects of drugs in unconditioned motor paradigms because the definition of behavioral categories depends idiosyncratically on the specific measuring device and testing environment. Second, correlations between large numbers of different behavioral categories pose difficult statistical problems and may not provide additional information about the behavior of the animal. Third, the behavioral effects of some drugs, such as the prototypical psychostimulant d-amphetamine, are characterized by a fragmentation of behavioral responses (Lyon and Robbins 1975), making it difficult to assess whether a specific behavioral category has been completed. Fourth, the determination of the neurobiological substrates underlying behavioral effects is complicated by the fact that a priori defined behavioral categories may not correspond to distinct neurochemical processes or neuroanatomical systems. Fifth, behavioral categories are not easily monitored with the temporal resolution that is necessary to detect subtle changes in the response topography of the animal. Alternatively, behavioral patterns can be extracted from measures describing distinct characteristics of behavioral events. Consequently, behavioral patterns can be defined as particular combinations of measure sequences observed within a collection of behavioral events. Several approaches have been developed to extract information about patterns of m o t o r activity in rodents. Stahle (1987) suggested that the extraction of linear factors based on a partial least square analysis relating measurable behavioral observables to pharmacological effects provides a superior assessment of behavioral patterns compared to multiple statistical comparisons of individual measures. Kernan et al. (1988) have used temporal correlation functions based on statistics of spatial point processes to detect pattern similarities and explore the behavioral repertoire of rats in a spontaneous m o t o r activity paradigm. Teitelbaum et al. (1982) have used time frame analysis to decompose behavioral responses into elementary events and assess their individual contributions. These approaches have provided additional and important information about m o t o r activity patterns. However, common shortcomings of these methods include their dependence on both predefined behavioral categories and a particular measuring resolution. Recently, we introduced scaling measures quantifying the functional relationship between an experimental parameter and a variable of interest describing a characteristic feature of the experimental system. These measures assess the relationship between individual behavioral events and thus quantify behavioral patterns of rats exposed to the Behavioral Pattern Monitor (BPM) (Paulus and Geyer 1991a). The computation of these measures is based on sequences of micro-events, defined by the smallest resolvable change in the event space. Using the scaling approach, these measures summarize the information contained on different spatio-temporal scales. Temporally, the contribution of short-duration micro-events in relation to long-duration micro-events is quantified by the temporal scaling exponent c~. Spatially, the average pathlength between microoevents separated by k microevents is quantified by the spatial scaling exponent d (Paulus and Geyer 1991a). This approach assumed that

the average scaling relationship adequately reflects the animal's behavior and provided no detailed information about the fluctuation of the scaling relationship. In this investigation, the scaling approach was extended to assess the geometrical characteristics of microevent subsequences. The spectrum of local spatial scaling exponents, f(d), describes functionally the contribution of micro~event subsequences with different local spatial scaling exponents, di. Thus, this function quantifies how sequences of behavioral events form macroscopic behavioral patterns. This paper focuses on three aims: I) to introduce the computational methods required to obtain the f(d) function; 2) to compare the relationship between the average scaling behavior and the f(d) function; and 3) to supply possible behavioral interpretations of changes in the f(d) based on an exemplary data set. A companion paper details the application of this method to analyze differences between classical dopaminergic psychostimulants and relatively selective dopamine reuptake inhibitors (Paulus et al. 1993).

Materials and methods

Basic scaling assumption. The scaling approach is based on the assumption that the relationship between experimental parameters and variables of interest provides important information about the structural and sequential characteristics of sequences. In general, behavioral observables are given by a set of coordinates in a predefined event space. Typically, discrete sets of coordinates are used to obain variables of interest. These variables of interest are collected for specific experimental conditions to yield measures of the observed behavior. For example, the (x, y) positions of a rat in a test environment and the time spent at that position provide a set of coordinates defining a micro-event of a behavior in a threedimensional event space. The variable of interest to determine the geometrical characteristics of path patterns is the distance or pathlength between two micro-events that are separated by k microevents. For example, the distance between consecutive micro-events, i.e. k = 1, is always 1 by definition of the micro-event as the smallest resolvable change in the event space. Here, k corresponds to the experimental parameter that is used to extract the structural information and is also called the resolution with which the pathlength is obtained. Specifically, the geometrical structure of movement sequences is extracted from the functional relationship between k and the pathlength. The particular functional forms relating these quantities are derived from scaling approaches in physical systems, which have yielded a remarkable universality (Stanley 1987; Barnsley 1988; Feder 1988), as well as from the description of a geometrical object using its fractal dimension (Mandelbrot 1977). This functional form can be stated most generally as [variable of interest] ([parameter A]) ~ [parameter A] '~"li~g0xp..... x [variable of interest] x ([parameter B]) Specifically, the average distance between micro-events separated by three micro-events, (L(3)), is related to the average distance separated by one micro-event, (L(1)) via (L(3)) ~ 32-d(L(1)) The scaling relationship is described quantitatively by the spatial scaling exponent, d. For a straight line, the distance after three micro-events (e.g. L(3)= 3) is three times the distance after one micro-event, whieh by definition is one (e_g.L(1) -- I), ie, 3 ,= 32 -d t. The value of d that solves this equation is 1. In contrast, a path pattern of tour micro-events characterized by two 90 degree right

179 turns will yield the same distance for two micro-events separated by three micro-events (e.g. L ( 3 ) = 1) as for two consecutive microevents (e.g. L(1) = 1). i.e. 1 = 32-al. The value of d that solves this equation is 2. In general, straight movements are associated with a low spatial scaling exponent, d, highly circumscribed movements with many turns and reversals result in a high spatial scaling exponent (Paulus and Geyer 1991a). The temporal scaling exponent, c~, relates the number of microevents to the micro-event duration, c~quantifies the amount of motor activity based on a local "degree of acting" by evaluating the ratio of short duration micro-events (high local degree of acting) versus long duration micro-events (low local degree of acting), d and e can differentiate unconditioned motor effects of pharmacological agents even when these substances result in similar levels of activity.

Extended scaling assumption. Patterns of behavior are not quantified sufficiently by average measures. The extended scaling assumption is based on the idea that the local scaling relationship between the variable of interest and the experimental parameter is characterized by a range of scaling exponents, d~, instead of only the average scaling exponent, d. The term local refers to a spatially or temporally limiting condition, i.e. the geometrical characteristics are calculated for circumscribed micro-event segments corresponding to consecutive movements of the animal. This idea reflects the experimental observation that animals display a variety of different movement sequences. The objective of this approach is to quantify how frequently specific d~ describe the scaling relationship of micro-event segments. To quantify the contributions of movements havirig different geometrical charactersitics in a manner that is independent of the resolution k, a function is defined that describes the relationship between the occurrence probability of specific dr and the resolution k. Specifically, the number of segments with a specific dr for k = 3, N(d~, 3), is related to the numl~er of N(di, 1) by the same functional form described above, i.e.

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Fig, 1. This figure schematically displays the construction of the function describing the spectrum of local spatial scaling exponents, f(d). Details are described in the text

N(d~, 3) ~ 3s~0 The resolution-independence of this function is an important feature due to the fact that the occurrence probability of specific d~ changes for different values of k. This scaling function has been developed to describe quantatively objects with diverse and complex geometrical characteristics (Halsey et al. 1986) and is called the spectrum of local

spatial scatin9 exponents, f(d). Graphical example. A schematic example should clarify the basic steps of calculating the spectrum of local spatial scaling exponents, f(d). In Fig. 1, four different segments constructed from five microevents at different levels of resolution are considered. The calculation of the pathlengths of these segments for k = 1 and k = 4 yields d~ = 1 for the straight line, de ~ 1.2 for the segment with triangular deflection, de = 1.5 for the bent segment, and d~ = 2 for the starshaped segment. For illustrative purposes, a path pattern is constructed from these segments on three levels of resolution, I, II, and Ill, corresponding approximately to k = 1,2 and 4, respectively. Based on the definition off(d) given above and the frequency of the different segments at k = 1 (1 for all segments) and k = 4 (3, 4, 8, and I4), thef(d) values for the four different segments are 0.79, 1.0, 1.5, and t.90, respectively. The f(d) function for the four different segments reads (1,0.79), (1.2, 1), (1.5, 1.5), and (2.0, 1.9). However, the path pattern also consists of combinations of these segments, which yields the functional extension between these individual points. Local spatial scalin9 exponents and the spectrum of local scalin9 exponents. For the analysis of rat movements, the Euclidean distance, A, between two micro-events is calculated for different segments, i, at different levels of resolution, k. In addition, the distances for different k are averaged within each segment to improve the estimation of de. The segment size was chosen as a compromise between two competing objectives. The first objective was to differentiate optimally the geometrical characterisitics of different movements, which required the use of small segment sizes. The second objective was to obtain an accurate least squares fit for different d~, which improves with large segment sizes. Extensive variations of

these parameters resulted in the selection of a segment size of 16 micro-events and the resolution range, k = 1-12. In addition, the contribution of artifactually small distances between consecutive micro-events due to the lower resolution limit of the recording instrument is reduced by "data dithering". This technique uses a small random number, ~, with magnitude less than 6 = 21 of the smallest micro-event distance, that is added to the (x, y) position to obtain the pathlength based on measuring the distance between two small areas of sidelength 2 6. Three micro-event segments consisting of 16 micro-events based on actual rat movements after "data dithering" are shown in Fig. 2. Three important conclusions can be drawn from the calculation of dl for these segments using k = 1 to k = 12. First, the log-log plots of the pathlength versus k supports the specific functional form of the scaling hypothesis described above. Second, the standard deviations of the d~s (about 1% of the di value) show that local spatial scaling exponents can be estimated reliably and accurately from these micro-event segments. Third, the large range of dr (t. 19-1.92) shows that the average d does not sufficiently describe the different path patterns and supports the extended scaling assumption. Despite the same amount of activity (counts = 16 micro-events), the geometrical characteristics of these segments differ drastically. While the movements in the first segment (Fig. 2, top left) are highly localized along the center of the short wall in the BPM chamber, the second segment shows a meandering pattern (Fig. 2, middle left), and the pattern of the last segment (Fig. 2, bottom left) is characterized by straight movements along the perimeter of the BPM chamber, Behaviorally, small dl values are associated with directed movements from one part of the enclosure to another, often serving diversive exploration (Berlyne 1960), but can also result from perseverative straight movements along the wall (Paulus and Geyer 1991 a). High local d~ values are obtained from animals engaging extensively in local behaviors, e.g. grooming, rearing, or holepoking, but can also occur for animals displaying focal stereotypic behavior, e.g. repeated hotepoking, gnawing, or head bobbing.

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Calculation of local spatial scaling exponents and the spectrum of local spatial scaling exponents. In the graphical example in Fig. 1, thef(d) function was estimated directly from the slope of the doubly logarithmic plot of the occurrence frequency of micro-event segments with a given d~ versus the measuring resolutions, k. In practice, however, direct estimations of thef(d) function depend sensitively on the range of the local spatial scaling exponent, ~d;, used to estimate their occurrence frequency. For practical calculations, a different method has been proposed (Halsey et al. 1986) that is based on the formal analogy of this functional formalism with statistical mechanics and thermodynamics (Chandler 1987). This analogy has been developed for fractal objects (Feder !988) and for dynamical systems (Ruelle 1989) and is summarized comprehensively in Badii (1989). In statistical physics, ensemble methods based on partition functions are used to obtain relevant measures. Specifically, a partition function, Z(q, n), is defined based on a "filtered summation" of the pathlengths of the micro-event segments. The behavior of Z(q, n) for different values of the filter parameter, q, is used to calculate both the local spatial scaling exponents and the fluctuation spectrum of local spatial scaling exponents (for details, see Appendix). Both di and f(di) depend implicitly on the filter parameter, q, since both descriptors are obtained from Z (q, n). First, micro-event segments are obtained from the sequences of micro-events generated by the animal during the exposure in the BPM. Second, three quantities, dq,f(dq), and Z (q, n), are calculated for a range of q values for these segments. The range of q values is selected to allow the calculation of a sufficient range for thef(d) function, such that the contributions of both low and high local spatial scaling exponents are determined. Typical q values range from - 4 to + 6. Specifically, the average spatial scaling exponent, d, previously used as a global descriptor of the geometrical path structure (Paulus and Geyer 199ia), is recovered for q = I, and dl = f(dt). Graphically, the d value can be located at the tangent of thef(d) function with the identity line (x = y). Third, the dq, f(dq), and Z(q, n) caIculated for each rat are averaged for all animals of a treatment group to obtain the group averages and the group standard deviation dq, a(dq), Z(q,n), a(Z(q,n)),f(dq) and a(f(dq)), respectively. For example, for each q value

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These group functions can be used to quantify the effects of treatments on the geometrical path structure.

Statistical comparisons between group averages of d and f(d). A de-

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tailed.quantitative assessment of drug effects on the path structure is obtained from the comparisons of d[q] and the f(d[q]) functions relative to the control group. Preliminary results indicate that the distribution of spatial scaling exponents in a large group of control animals does not deviate significantly from a normal distribution (Paulus and Geyer 1993). Thus, the significance of a drug effect can be expressed by a z-transformed normal score. However, some drug treatments may change the variance of the average spatial scaling exponent. Therefore, a modified z-transformation was used to obtain a conservative estimate of the significance of treatment effects. This transformation includes the standard deviations of the control group, ac(f(d)), and of the treatment group, crA(f(d)). The difference curves, f~. ~(d) are called local pattern differences and signify" alterations of the contributions of specific dzs relative to controls, but do

15 20

log(k)

Fig. 2. This figure displays three segments of 16 consecutive microevents after "data dithering" for a rat in the BPM. Each micro-event corresponds to a point, with successive micro-events being connec-

ted by lines. The local spatial scaling exponent, d, and its standard deviation (SD) are shown for each path segment. The corresponding plots of the logarithm of the pathlength versus the logarithm of the resolution are displayed on the right of each segment. The straight line signifies the result of a least squares fitting procedure; the slope determines the local spatial scaling exponent

181 not indicate overall changes in the path structure. These changes are assessed most easily by z-transformed differencesbetween the average spatial scaling exponents, d~ (Paulus and Geyer 1991a). These overall changes are displayed comprehensively in the d - e plane, which simultaneously describes changes in both the temporal scaling exponent, cq and the spatial scaling exponent, d. Therefore, both the z-transformed d-differences and the local pattern differences yield specific and different information about the geometrical path structure. While the d~ indicates the global changes in the geometrical path structure, theft. A(d) quantifies local changes in the movement sequences.

Rat locomotorbehaviorassessment.The Behavioral Pattern Monitor and general experimental procedures are described in the companion paper (Paulus et al. 1993). Briefly,for each experiment, animals received subcutaneous injections of drugs 10 min prior to testing. Typically, animals were exposed to the BPM for 120 rain, but only the first 60 min were used to calculate the f(d) functions shown below. Detailed descriptions of the drug treatments for the RU 24969 animal can be found in Rempel et at (1993), for the 2,5dimethoxy-4-iodo-amphetamine (DOI) animal in Paulus and Geyer (1992), and for the quinpirole animal in Paulus and Geyer (1991b).

. Results

Examples of different f(d) functions Figure 3 displays the results of five exemplary animals treated with saline, the dopamine (DA) D-2 agonist quinpirole (0.0625 mg/kg), the DA uptake inhibitor nomifensine (5.0mg/kg), the serotonin 5-HTz agonist DOI (0.27mg/kg), and the 5-HT1B agonist RU 24969 (1.25 mg/kg), respectively. These selected animals exemplify combinations of the average spatial scaling exponent, d, and the temporal scaling exponent, e, in all four quadrants of the d - ~ plane. The d - e plane illustrates differential effects of drugs on the amount and average path structure of unconditioned motor activity (Paulus and Geyer 1991a). The symbols in the schematicized d - c~ plane indicate the approximate location of the drug treated animals. The saline control animal is located at the origin. The locomotor paths of individual animals are displayed by the lines connecting consecutive (x, y) locations in the BPM. The resolution limit of this instrument, a grid of 7 x 15 possible locations, is evident in the graph's square patterns. The paths are displayed for 60 min of unconditioned motor activity in the BPM chamber. Instead of plotting the exact (x, y) location of the animals, the displayed (x, y) position is obtained by choosing randomly from a range of values (x _+ 0.4, y _+ 0.4) surrounding the exact location. This procedure avoids retracing the same lines and thereby facilitates the visual recognition of frequently visited areas. The path structure of the saline control animal is characterized by frequently visited locations, which have been called the home area (Geyer t990), as well as preferred paths throughout the BPM chamber. Straight movements can be found along the long walls of the BPM enclosure, while local movements are observed particularly along the short walls and in the corners of the chamber. While the quinpirole- and nomifensine-treated animals differ vastly in the amount of motor activity

exhibited (1705 and 9745 micro-event counts during the first 60 rain, respectively), the path patterns share some similarities. Specifically, local areas of increased movement density reflect frequent local movement sequences characterized by high local di values. In addition, occasional long straight movements can be observed along the walls of the BPM chamber. Similarly, the DOI- and RU 24969-treated animals are distinguished by their amount of motor activity (3793 and 10316 micro-event counts during the first 60 min, respectively). The movement patterns of both animals, however, are dominated by long straight paths primarily along the walls of the BPM chamber. Correspondingly, movements into the center area of the enclosure are infrequent. The overall path structure for quinpirole- and nomifensine-treated animals yield similar average d values, of 1.71 and 1.70 respectively. In contrast, the path structure of DOI- and RU 24969-treated animals resulted in average d values of 1.48 and 1.49, respectively. The average d, therefore, distinguishes the different path structures of drugs that either reduce or increase unconditioned motor activity. However, closer examination of these path structures also discloses differences both between quinpirole and nomifensine and between DOI and RU 24969. For example, locations of highly circumscribed movements set apart the path structure of the nomifensine animal from the path structure of the quinpirole animal. While this descriptive assessment of these subtle path characteristics is complicated by the confounding influence of the difference in the amount of motor activity, the spectrum of spatial scaling exponents, f(d), enables a quantification of the path structure independent of the amount of motor activity. The f(d) functions for all five animals are displayed with the corresponding symbols at the points where the function has been evaluated using the q-dependent mass scaling exponent. Typically, these functions range from 1.0 _< di < 1.5, describing extremely straight or directed movements, to 2.0 _< dl < 2.5, reflecting highly circumscribed movements. The f(d) function for the saline control animal is shown without a symbol. While the quinpirole and nomifensine treatments shift thef(d) function to the right, DOI and RU 24969 treatments shift the f(d) function to the left. In both cases, the left ascending branch of thef(d) functions of low di values overlap. Thus, the quinpirole- and nomifensine-treated rats are similar with regard to the contributions of micro-event segments with straight movements. Similarly, the relative contributions of straight movements does not differ for the DOIand RU 24969 treatments. However, the right descending branch of the f(d) function reveals that the nomifensinetreated animal exhibited an increased contribution of highly local movements compared to the quinpiroletreated animal. Highly local movements also contribute more to the overall path structure for the DOI animal relative to the RU 24969 animal. In both cases, these local differences are well within the global shifts of the f(d) functions. These differences within the overall shift of the functions indicate that, although relative differences exist between quinpirole and nomifensine or DOI and RU 24969, these changes are below and above the respective contributions of di values for the saline animal.

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Comparison of the f(d) functions of two saline control groups

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Figure 4a shows the groupf(d) functions for two different saline control groups, each consisting of eight animals. The SEM for f(d) is shown where the function has been evaluated from the mass scaling exponent. The d~ values contributing significantly to the overall path structure for both functions range between 1.25 and 2.2. The average d measures evaluated at the tangent point with the diagonal identity line were found to be 1.629 and 1.627, respectively. While the left ascending branches of both f(d) functions overlap, the right descending branches reveal small differences between these control groups. The evaluation of the local pattern differences (Fig. 4b),f~ (d), shows, however, that these differences are well within the two range of the averaged variance. In addition, the local pattern differences are negligible throughout the scaling range from highly straight path patterns to fairly local movements. Thus, thef(d) functions do not differ significantly within the tested scaling range. Similar results have been obtained with several other control groups across a variety of experiments.

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Discussion The results of the exemplary data sets demonstrate that the evaluation of the geometrical characteristics of movements using thef(d) function has three important features. First, thef(d) function assesses the path patterns independently of the amount of activity exhibited during the observation period. The geometrical characteristics of the rat movements were similar for quinpirole and nomifensine or DOI and RU 24969, despite the fact that quinpirole and DOI reduced the amount of motor activity while nomifensine and RU 24969 increased the number of micro-events. Second, the average spatial scaling exponent, d, reflects average changes in the path patterns, corresponding to shifts of the f(d) function, but is not sufficient to describe the detailed changes in the geometrical structure of movements. While both quinpirole and nomifensine or DOI and RU 24969 shared similar average d values, thef(d) functions differed distinctly in the range of local spatial scaling exponents quantifying highly circumscribed movements. Third, changes of the f(d) function can occur in both directions of the scaling range. Specifically, in exemplary animals, relatively large shifts to the right were induced by quinpirote or nomifensine, whereas opposite left shifts occurred after treatment with DOI or RU 24969. A more detailed analysis of drug effects is provided in the companion paper (Paulus et at. 1993).

Fig. 3. This figure depicts the results of the f(d) evaluation for a saline-treated rat and representative animals treated with quinpirole (0.0625 mg/kg), nomifensine (5.0 mg/kg), DOI (0.27 mg/kg), or RU 24969 (1.25 mg/kg), respectively. Thef(d) functions are displayed on the top right with the symbols corresponding, to the symbols indicating the approximate location of these drugs in the d-:~ plane (center): The rat paths are graphed for the first 60 min of activity in the BPM chamber

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Fig. 4. This figure displays thef(d) evaluation (top)for two different control groups consisting of eight_animals each. The SEM is indicated for each point along the f(d) function. The local pattern differences,f~(d) (bottom), show the differences of the geometrical characteristics of the movements between these control 'groups. These differences are non-significant for the entire scaling range from local d = 1.0 to local d = 2.3

These large shifts are particularly intriguing with respect to two important characteristics of the grotip f(d) functions. First, the deviations in the contributions of different local dl values to the overall behavior, i.e. thef(d) values for different animals were extremely small. The SEM of the points along thef(d) functions were generally several orders of magnitude smaller than the changes observed for the exemplary animals. Second, control groups from different experiments shared extremely similar f(d) functions. The assessment of pattern differences revealed that the path structure of both control groups did

184 not differ significantly. How_ever, the variation along the descending branch of the f(d) function appears to be larger than along the ascending branch. To summarize, thef(d) function provides a stable, sensitive, and detailed quantitative assessment of the geometrical structure of path patterns independent of the amount of motor activity. While this quantitative descriptor of the geometrical structure of path patterns appears to have important features required for a proper assessment of motor behavior (Robbins 1977; Reiter and MacPhail 11979; Geyer 1990), the question remains whether an additional assessment of path patterns is necessary to characterize unconditioned responses to pharmacological agents. In a recent assessment, Robbins and Everitt (1990) emphasized that dopaminergic, cholinergic, and noradrenergic neurotransmitter systems differentiallymodulate responses used as indices of arousal and attention. Furthermore, it has been proposed that the cholinergic nucleus basalis as well as serotonergic systems play key roles in neocortical activation and arousal (Bauman et al. 1989; Buzsaki and Gage 1989). Consistent with these hypotheses, catechotamine releasers, serotonin releasers and 5-HTIB agonists such as RU 24969 as well as cholinergic drugs such as scopolamine can profoundly increase levels of motor activity (Geyer et al. 1986; Callaway et al. 1990; Rempel et al. 1993). Recently, we found that dramatic changes in the geometrical structure of rat movements can distinguish these pharmacologically diverse drugs (Paulus and Geyer 1991a). For example, the average spatial scaling exponent, d, clearly distinguished the motor activating effects of d-amphetamine, methylenedioxymethamphetamine (MDMA), and scopolamine. While d-amphetamine did not change the average scaling behavior, both MDMA and scopolamine significantly decreased d, indicating a shift towards straight path patterns (Paulus and Geyer 1991a; 1992). The geometrical structure of unconditioned motor behavior appears to be an important aspect of the different types of motor activation associated with arousal induced by different pharmacological agents. However, pharmacological substances frequently activate different neutrotransmitter systems simultaneously or affect the same system at different neuroanatomical locations. The f(d) function attempts to quantify selectively the contributions of different motor behaviors and therefore provides a differential description of the arousal state. This function may be useful for determining the contributions of different neurotransmitter systems to the behavioral effects of drugs. From a behavioral perspective, the f(d) function precisely quantifies different geometrical characteristics of motor behavior. This function describes relationships between microscopically recorded variables, in this paradigm consecutive (x, y) locations, and macroscopically observed behavior, i.e. different patterns of motor behavior. Typically, behavioral patterns are defined a priori and labeled as categories such as exploratory behavior, grooming, rearing, focal stereotypies, etc. By contrast, the f(d) function is obtained without imposing a priori distinctions between behavioral categories and instead describes contributions from the entire range of local d~ values. While particular ranges of local d~ values may

correspond to such categories, no such assumption is made here. Furthermore, instead of calculating an average descriptor of the geometrical characteristics of motor behavior, the f(d) function quantifies the relative contributions of different movement sequences to the average path pattern~ The scaling assumption provides a method to extract the geometrical characteristics of unconditioned motor behavior that eliminates the dependence on the measuring equipment and the precise definition of a micro-event. Basically, three different types of information can be obtained in unconditioned motor paradigms. First, the general level of activity can be measured via photobeams, capacitance plates, or video systems, indicating the general level of the animal's arousal(Robbins 1977; Geyer 1990). Second, specific behavioral responses can be obtained if the enclosure is equipped with holes or objects that can be explored (Ljungberg and Ungerstedt 1977; Geyer et aL 198@ Third, the patterns of movement sequences can be assessed with respect to their dynamical features, i.e. their predictability (Paulus et al. 1990, 1991) and their geometrical or spatio-temporal characteristics (Geyer et al. 1986; Paulus and Geyer 1991a). While the first evaluation can be obtained from discrete microevents of behavior, the second evaluation requires an a priori definition of distinct behavioral categories. The third assessment, however, relies solely on the relationship between measures from different micro-events. The geometrical information obtained from the spectrum of spatial scaling exponents, f(d), is based on the relationship of distances between micro-events and the separation between these micro~eventso This type of information requires knowledge about the location of consecutive micro-events. Thus, this assessment provides information that cannot be obtained simply by quantifying aspects of single micro-events. Instead, the relationship between individual distances or movements, which is typically used to define a priori behavioral categories, determines the shape of the f(d) function. Typically, response topography refers to the contribution of different behaviors that are distinguished by characteristic movement sequences. Consequently, the f(d) function enables a quantification of the animal's response topography, based on the geometrical characteristics of movements. The f(d) function is based on the assumption that variables obtained from the relation between different micro-events, such as the distance between k micro-events, can be described by a scale-invariant functional relationship. The specific functional form of the scaling equation has been used extensively in the description of physical systems such as fractal aggregates, river levels, etc. (Feder 1989). The resulting scaling exponents have classified these systems into distinct universality classes, thereby providing a precise fingerprint of the behavior of these physical systems. The fluctuation between different local scaling exponents is assumed to depend predominantly on the value of the local scaling exponents and not on the length scale used to measure the contribution of the scaling exponent. This assumption can be tested explicitly by evaluating thef(d) function via a different approach using the direct estimation of the changes of the occurrence frequency with changing measuring resolution. However,

185 since this a p p r o a c h requires larger d a t a sets, it m a y be m o s t a p p l i c a b l e to c o m b i n e d p a t h p a t t e r n s of m u l t i p l e animals, which m a y be a d i r e c t i o n for future investigations. T h e m a i n difference between the a p p l i c a t i o n s of scaling m e t h o d s in physical ( F e d e r 1989) o r m a t h e m a t i c a l ( M a n d e l b r o t 1977; Barnsley 1988) systems a n d biological systems is the existence of a limited scaling range in the latter case. Specifically, the d e s c r i p t i o n of p a t h p a t t e r n s in the B P M was c a l c u l a t e d from a scaling range of 1-16 micro-events. Therefore, these p a t t e r n s are n o t fractal objects in a rigorous sense. Nevertheless, within the specified scaling range, they have features similar to fractal objects a n d can be quantified p r o p e r l y b y a fractal-like a p p r o a c h . T o s u m m a r i z e , the e x t e n d e d scaling a p p r o a c h using the s p e c t r u m of s p a t i a l scaling exponents, f ( d ) , p r o v i d e s a fully a u t o m a t e d , r e s o l u t i o n - i n d e p e n d e n t q u a n t i f i c a t i o n of g e o m e t r i c p a t h p a t t e r n s with a large range of variations, a stable c o n t r o l g r o u p condition, a n d differential assessment of different local m o v e m e n t sequences. This function is n o t d e p e n d e n t on a priori b e h a v i o r a l categories b u t m a y be used to correlate different b e h a v i o r a l categories with their respective local spatial scaling exponents. Acknowledgements. This work was supported by a grant from the National Institute on Drug Abuse (DA06325). MA Geyer was supported by a Research Scientist Development Award from the National Institute of Mental Health (MH00188). We thank Diana Martinez, Marcus Chacon, Virginia Masten, and Richard Sharp for their assistance in the conduct of these studies and Clif Callaway for valuable discussions clarifying the presentation of this evaluation technique.

Appendix The pathlength or distance, L(i, n, k), of the ith micro-event segment of length, n, based on Euclidean distance, A, between two rat positions, (x,y)j and (x, Y)j+k measured with resolution k is calculated from i+n-k

L(i,n,k) =

J=

N-k

~ log ( L ( i , n , k ) ) L ( i , n , k ) q-1

-- ~Z(q,n)

i=0

dq = lim k~o

(?q

N-k log (k) ~ L(i,n,k) q-1 i=0

and the spectrum of local spatial scaling exponents,f (d), is calculated from the following relationship f(dq) = dqq - Z(q, n)

which is known as Legendre transform in statistical mechanics and thermodynamics (Feder 1988). The q parameter is analogous to the temperature in physical systems and to an undetermined or Lagrange multiplier in mathematical systems. An immediate interpretation of dq and rid[q]) can be obtained for q = 1. In this case, the average spatial scaling exponent, d, is obtained N-k

log (L(i,n,k)) ( log(L(n, k))) dl = lim i-o _ = d k~O log(k) (N - k) log(k) Moreover, Z(1, n) = O, therefore f(d) = d

The mean and the standard deviation of the groupf(d) function are calculated from f(dq) = d~q - Z(q,n)

and f(dq) + a(f(dq)) = (d + a(dq))q - (Z(q,n) + a(Z(q,n))) f(dq) - a(f(d~)) = (d~ - a(dq))q - (Z(q, n) - a(Z(q, n)))

To determine the effects of different treatments on the detailed geometrical characteristics of the path patterns, local pattern differences, f~(d), are calculated as modified z-transformations between the control group spectrum of local spatial scaling exponent, fc(d) and the treatment group function, fA (d). Specifically, f~, a (d) =

~,(d) - J~ (d) [ac(f(d)) + aA(f(d))] 2

here ac(f(d)) and aA(f(d)) correspond to the standard deviation of the control group and treatment group, respectively.

k 2 x (n - k)

Here, L(i, n, k) is corrected by a factor of k2 to simplify the local scaling relationship between k and L(i, n, k), which reads L(i,n,k) ~ k -'~"''~

For all calculations n = t6 and k was varied between t and 12. The first 16 micro-events yield L(1,16, k) and the pathlength for the last micro-event segment is given by L ( N - t6, 16, k) where N denotes the total number of micro-events. The partition function, Z(q, n), for the pathlengths of the microevent segments is defined by N-k

k z(q,.)~ ~ L(i,n,k)q-~ i=o

and has been called the mass scaling exponent. Algorithmically, it is obtained from the least squares fit of Z(q, n) for a range of k, i.e.

Z(q, n) ~ --

from the rate of change of Z(q, n) with respect to q, i.e.

log(k)

The q-dependent local spatial scaling exponents, d~, are obtained

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