## Abstract

Studies of human navigation in virtual maze environments have consistently linked advanced age with greater distance traveled between the start and the goal and longer duration of the search. Observations of search path geometry suggest that routes taken by older adults may be unnecessarily complex and that excessive path complexity may be an indicator of cognitive difficulties experienced by older navigators. In a sample of healthy adults, we quantify search path complexity in a virtual Morris water maze with a novel method based on fractal dimensionality. In a two-level hierarchical linear model, we estimated improvement in navigation performance across trials by a decline in route length, shortening of search time, and reduction in fractal dimensionality of the path. While replicating commonly reported age and sex differences in time and distance indices, a reduction in fractal dimension of the path accounted for improvement across trials, independent of age or sex. The volumes of brain regions associated with the establishment of cognitive maps (parahippocampal gyrus and hippocampus) were related to path dimensionality, but not to the total distance and time. Thus, fractal dimensionality of a navigational path may present a useful complementary method of quantifying performance in navigation.

## Introduction

For a wide range of organisms, the ability to find a way in a complex environment is arguably a prerequisite for survival (Wolbers and Hegarty 2010). The Morris water maze (MWM; Morris 1981), one of the most popular laboratory tasks in studies of animal navigational behavior, reliably demonstrates age-related deficits in spatial navigation across species and has been recognized as an excellent candidate for a universal test of spatial navigation skills. In a typical MWM experiment, an animal (usually a rodent) is placed in a wide cylindrical container filled with opaque water and is left to navigate toward a hidden, stationary platform using landmarks positioned around the water tank. In a virtual adaptation of this task (vMWM), a human participant navigates a virtual “water-filled” basin that includes the characteristic hidden, stationary platform and landmark cues. Navigational performance in a MWM is sensitive to a multitude of environmental, neurosurgical, pharmacological, and behavioral manipulations and after its introduction (Morris 1981), it became a very popular research tool: typing “Morris Water Maze” in a popular search engine yields more than 5000 entries (http://www.ncbi.nlm.nih.gov/pubmed/?Term=Morris+Water+Maze, accessed on 13 January 2014).

Because age-related deficits in spatial navigation are observed across species (Gallagher et al. 1993; Lindner 1997; see Moffat 2009 for a review), the traditional and virtual MWM provide an opportunity for valid comparative studies and hypotheses testing. Indeed, the extant literature indicates that similar to rodents swimming in an MWM (Gallagher et al. 1993; Lindner 1997), older humans cover longer distance and require more time to reach the platform than their younger counterparts do (Kirasic 1991; Kirasic et al. 1992; Moffat and Resnick 2002; Driscoll et al. 2003, 2005; Moffat et al. 2007). In comparison to younger animals, aged rats require more trials to reach an asymptotic level of performance but still demonstrate a deficit in later test trials (Rapp et al. 1987). These persistent age-related deficits are frequently observed but poorly understood. They have been attributed to poor temporal-spatial integration of landmark cues (Rapp et al. 1987; Kirasic et al. 1992) and decline in spatial working memory (Magnusson et al. 2003). The complex reasons for age-related deficits in navigation are not easily captured by aggregated indices such as total time of search and distance from start to the platform.

Even a cursory look at the paths traveled by the participants navigating the vMWM suggests that less efficient performers tend to take less direct and more tortuous routes to the goal. In search of the platform, successive movements reflect decisions that are based on individual and environmental factors. When variations in the environmental conditions are controlled, individual differences in the complexity of a search path may reflect differences in cognitive processing. When compared with their young counterparts, older adults seem to travel exceedingly complex paths. These observations led to a hypothesis that, unlike their younger peers, older participants do not form reliable spatial representations that can be manipulated and updated in the process of navigation (Kirasic 1991; Kirasic et al. 1992; Moffat and Resnick 2002; see Moffat 2009 for a review). The internal spatial representation, a cognitive map (O'Keefe and Nadel 1978), is referenced while navigating an environment. It may be crucial for establishing one's location and predicting success in reaching a target following a specific sequence of movements (O'Keefe 1990). The decisions made en route to an intended target determine the geometry of the search path, which is believed to reflect the individual's cognitive map. When the observed search path is straight-forward and goal-oriented, the inference is that the participant is able to form and use a detailed and stable cognitive representation of the environment. As navigation performance improves, the geometry of the navigation path is expected to change, presumably reflecting adjustments to the cognitive map.

In spite of the theoretical importance of explaining individual differences in navigational performance, the path geometry is usually ignored and aggregate measures such as distance or time between start and attainment of the platform are used instead. To answer a question of potential influence of path geometry on navigation time, a common practice has been a visual inspection of individual paths (i.e., Sei et al. 1992; Moffat and Resnick 2002). This approach often relies on extreme examples and yields no quantitative indices amenable to statistical analyses. Categorical classification of search patterns only partly addresses the latter concern. Based on variable definitions, a search path may be classified as direct or indirect, spatial or nonspatial (Davis et al. 2010), or assigned to any category depending on the researcher's interest. Such an approach is subjective, and creating nominal classes produces little statistical variability, thus limiting options for analysis and interpretation. Counting the number of path crossings (Buzsaki 2005) is a third alternative that offers more statistical flexibility. A path crossing is defined as crossing a previously visited xy coordinate. The sum of path crossings is thought to reflect path tortuosity, with a larger number indicating compromised spatial cognition. Counting path crossings provides a continuous measure but ignores dynamic path changes by representing them as discrete events. Heading error is not an index of search path complexity but has been used to infer decision-making processes in the MWM based on navigation accuracy (i.e., Smith et al. 2013). In older navigators, heading error may be greatly influenced by turning biases, which may increase with age and are related to brain volume asymmetry (Yuan et al. 2013). Such individual and age-related biases render interpretation of heading error data rather difficult.

To address the outlined limitations and to find a suitable index of path complexity, we propose to treat a search path as a geometrical object and to use its fractal dimensionality as a novel measure of navigational performance. Since the introduction of fractal geometry (Mandelbrot 1967), FD has been applied to the study of many natural and behavioral phenomena, including patterns of animal migration and grazing (Garcia et al. 2005; Gautestad and Mysterud 2010; Gautestad 2011). Migration and foraging paths have a distinct commonality with patterns of human navigation (Hills et al. 2013), and application of FD to the latter may prove as useful as its use in the former.

In the observed segment of the environment, FD is an index of complexity or tortuosity of a movement path between 2 fixed points. By quantifying the complexity of a search path, statistical patterns may emerge that can elucidate the complex interplay of environmental and cognitive factors (Gautestad 2011). FD is a continuous measure of geometrical properties of an object confined to a plane (Mandelbrot 1967; Milne 1991). A straight line will have FD = 1, and a complex pattern filling the plane completely will have FD = 2. FD is scale-dependent and is calculated by the change in complexity as the measurement scale changes. The change in the logarithm of FD is a linear function of the change in the logarithm of scale with a slope equal to 1-FD (see Sugihara and May 1990). Fractal dimensionality does not assume limitless movement, and therefore, FD can be applied to calculate path complexity within a bounded space (Garcia et al. 2005), such as in the MWM pool. Although FD to some extent depends on the total distance traveled, the measures are not redundant. Inspection of the examples in Figure 1 reveals that more circuitous paths that are associated with a greater distance may have a lower FD (e.g., paths A and B). Such paths may signify a directed albeit an inefficient search pattern. Increasing FD suggests an increasing randomness and uncertainty in the search pattern (Figure 1C). Thus, FD is a continuous measure that is sensitive to the total path dynamics and, therefore, may explain individual differences in spatial navigation that have to date escaped quantification by traditional measures of navigation performance (i.e., distance to the goal and time of travel).

Figure 1.

Three example paths are shown with calculated FD, path length (virtual units), and travel time (s). Example paths (A and B) show that low fractal dimensionality (approaching 1) can occur in long paths, whereas (C) demonstrates an increasingly random search pattern (FD approaching 2).

Figure 1.

Three example paths are shown with calculated FD, path length (virtual units), and travel time (s). Example paths (A and B) show that low fractal dimensionality (approaching 1) can occur in long paths, whereas (C) demonstrates an increasingly random search pattern (FD approaching 2).

To evaluate the potential of this measure, we calculated the FD of individual search paths in a vMWM and modeled the change in time, distance, and FD across 6 learning trials. The vMWM is an open-field exploration with unrestricted search and thus provides a promising environment to test the relative utility of FD in comparison to other measures of navigation performance. Evaluation of multiple indices of navigational performance can lead to insights into individual differences in travel time and distance that are currently de facto standards of assessment in vMWM navigation. We hypothesize that advanced age would be associated with poorer performance as measured by these 3 indices but that FD will provide information about age differences that is not captured by time of search and distance traveled to the platform.

## Methods

### Participants

The participants were 139 healthy community-dwelling volunteers aged 18–77 years (M = 48.52, SD = 15.85; see Fig. 2 for age distribution of the sample). They were part of a larger longitudinal study of neural correlates to cognitive aging and were screened for history of cardiovascular, psychiatric, and neurological disease, hypertension, diabetes, thyroid disorder, drug and alcohol abuse, sensory impairments, dementia, depression, and head trauma. For inclusion, participants had to score below 16 on the Geriatric Depression Questionnaire (CES-D; Radloff 1977) and above 25 on the Mini-Mental State Examination (Folstein et al. 1975). They were right-hand dominant (75% and above on the Edinburgh Handedness Questionnaire; Oldfield 1971) and were native English speaking. All participants were normotensive, that is, were not taking antihypertensive medication and had normal systolic and diastolic blood pressure (see below). Fourteen participants were excluded from the original sample due to diagnosed or observed hypertension. Forty-four women were post-menopausal, of which 5 received a hormone replacement therapy (treatment duration = 3–14 years); 8 pre-menopausal women reported taking hormonal contraceptives; no men reported a hormone replacement therapy. Table 1 presents the demographic profile of the sample.

Table 1

Sample description

Total Men Women t-test
N 139 47 92
Age (years) 48.52 ± 15.85 47.96 ± 17.53 48.80 ± 15.02 0.28
Education (years) 15.76 ± 2.35 15.53 ± 2.25 15.88 ± 2.39 0.84
MMSE 28.89 ± 1.08 28.81 ± 1.10 28.92 ± 1.08 0.59
CES-D 4.46 ± 3.97 5.15 ± 4.63 4.11 ± 3.57 −1.35
Total Men Women t-test
N 139 47 92
Age (years) 48.52 ± 15.85 47.96 ± 17.53 48.80 ± 15.02 0.28
Education (years) 15.76 ± 2.35 15.53 ± 2.25 15.88 ± 2.39 0.84
MMSE 28.89 ± 1.08 28.81 ± 1.10 28.92 ± 1.08 0.59
CES-D 4.46 ± 3.97 5.15 ± 4.63 4.11 ± 3.57 −1.35

Note: All t-test comparisons are nonsignificant at P >0.05.

Figure 2.

Age distribution of the sample.

Figure 2.

Age distribution of the sample.

### Testing Procedure

All cognitive measures were collected during testing sessions that lasted approximately 1.5–2 h and were conducted by trained technicians. Participants were confirmed to be normotensive according to the average of 3 blood pressure measurements, each collected at the start of a cognitive testing session by a mercury sphygmomanometer (BMS 12-S25) with a standard blood pressure cuff (Omron Professional) on the left arm while the participant was seated and resting the forearm on a table. Participants were dichotomously classified as hypertensive if the individual had a clinical diagnosis of hypertension, was currently taking prescription of antihypertensive medication, or if average blood pressure exceeded 140 (systolic) or 90 mm Hg (diastolic).

#### Pretest Training and Practice

We used a computerized vMWM task developed by Moffat and Resnick (2002). A detailed description of the virtual environment and testing procedures has been previously reported in Moffat et al. 2007. Briefly, prior to placement in the vMWM pool, participants were introduced to 2 unique virtual environments and trained to manipulate a joystick to travel through space. For practice in a vMWM, participants were placed in a circular pool that was in a room with notable wall geometry and objects around the pool perimeter. Each participant completed 6 practice trials by “swimming” toward a stationary platform that alternated between being hidden and visible on successive trials. Throughout the practice, participants were reminded that the environment remained the same and that the platform was stationary.

After practice, all participants completed 6 learning trials that followed the same format as the practice, except that the stationary platform was always “hidden.” The task was still performed in a circular pool, but the location of the platform and the environmental cues were new. Starting positions were in each of the 3 quadrants without the platform and were repeated twice. Measures of spatial navigation included time and distance traveled until the first platform intersection and path FD for each of the 6 place-navigation test trials. The search behavior was recorded as xy coordinates in the virtual pool, sampled at 100 Hz rate. The participant could control the speed of movement by manipulating the joystick. No time limits were imposed on the participants, and each trial ended upon successfully locating the platform, which resulted in platform animation lifting above the water plane and production of an audible signal. Following the place-navigation trials, participants performed a fixed 1-min probe trial in which there was no platform, but participants were instructed that the environment had not changed. FD is not an appropriate measure for a probe trial, as the measure is only meaningful for search paths toward a discrete end point that will terminate the search path. In the probe trial, crossing the platform location does not terminate the trial, and the individual will continue to explore the vMWM. Because FD is not a meaningful measure under these conditions, probe trial data are not reported here.

#### Control Measures

We employed 2 control measures to assess possible age- and sex-related differences in proficiency of joystick control and prior experience with 3D virtual environments (i.e., video games). Participants repeated the task in the same environment with a visible platform. Participants were instructed to move toward the visible platform as quickly as possible, and distance was treated as a control measure for individual differences in joystick control. Finally, participants completed a 3-item questionnaire about prior experience with computers and video games. Based on a 7-point Likert rating (1—Never; 7—Every day), participants reported how often they played video games that displayed a 3D environment.

### Calculation of Fractal Dimension

The FD of each search path was calculated from the xy coordinate output using Fractal (v 5.20) software developed by Nams (Dalhousie University, http://www.dal.ca/faculty/agriculture/environmental-sciences/faculty-staff/our-faculty/vilis-nams/fractal.html, accessed on 14 May 2014). All calculations were performed with the Fractal Mean function (freeing estimation parameters), which samples the path twice and corrects for truncation at the beginning and end of the path. FD was calculated by slope of the function defined by change in log(distance) and log(spatial scale) (Nams 2006).

### MRI Acquisition and Volumetric Measurement

Structural imaging was performed on a Bruker by Siemens 4-Tesla MRI system with an 8-channel head coil. A T1-weighted magnetization-prepared rapid gradient echo sequence was acquired with the following parameters: 0.67 × 0.67 × 1.34 mm3 voxel, echo time = 4.38 ms, repetition time = 1600 ms, inversion time = 800 ms, field of view = 256 mm, matrix size = 384 × 384, and flip angle = 8°. In post-processing, images were reformatted in native space to adjust for variation in head position with Analyze 8.0 software (Biomedical Imaging Resource, Mayo Clinic College of Medicine). All images were manually demarcated by trained raters who had confirmed reliability of all volumetric measures with an intraclass correlation coefficient of at least 0.90 (ICC(2), assumed random raters; Shrout and Fleiss 1979).

The volumes of 6 brain regions of interest (ROIs)—the Cd, Hc, PHG, Cb, primary visual cortex (VC), and the orbito-frontal and dorsal lateral prefrontal cortices (collectively PFC)—were manually measured on the MRI scans as described in previous publications (e.g., Raz et al. 2004; Raz et al. 2005; Raz et al. 2010). The left and right hemisphere volumes of every ROI were summed and adjusted for individual differences in intracranial volume using the ANCOVA approach (Jack et al. 1989; Raz et al. 2004).

### Data Conditioning and Statistical Analysis

#### Analyses of Behavior Models

The 3 starting positions within the vMWM were not equidistant from the platform. Therefore, to facilitate comparison across trials, distance and time measures were adjusted for the minimum required to reach the platform from a given starting position. To address moderate skew, time and distance measures were log-transformed and FD was winsorized. Average differences in the 3 navigation measures were estimated with general linear models (GLMs). Age (mean-centered) and sex differences were examined in a repeated measure GLM with performance indices computed for each participant forming a 3-level (time, distance, and FD) within-subject factor. Post-hoc hypothesis testing and univariate GLMs were used to follow up differential effects by performance index.

To measure the change in spatial navigation across trials, we fitted two-level hierarchical linear models (HLMs). In the first model, change across trials in maze performance indices (time, distance, and FD) was estimated at level 1, and the intercept and the slope of this change were evaluated as a function of age (mean-centered) and sex. The second HLM estimated the change in travel time (Y) across trials, predicted by distance traveled and FD at level 1. At level 2, the relationship between the intercept, slope of the regression of travel time on virtual distance and on FD across trials, as well as change due to repeated trial alone were estimated as functions of age (centered at the sample mean) and sex.

#### Brain and Behavior Models

Secondary to the behavioral analysis, we examined the relationship between brain volumetry and navigation performance. Regional brain volumes were entered as level 2 covariates to predict differences in intercept and change in time, distance, and FD. For the purposes of the analysis, brain volumes were winsorized to alleviate the influence of extreme observations and were transformed with a linear constant to equate volume scales across regions. To minimize the bias due to collinearity between regional brain measures (r = 0.21–0.47), volumes were entered as covariates into separate models (all measures were mean-centered) (see Appendix for all HLMs).

## Results

### Behavioral Analyses: Age, Sex, and Trial-by-Trial Differences

Advanced age was associated with longer search time (β = 0.23, P = 0.001), longer traveled distance (β = 0.50, P < 0.001), and larger FD (β = 0.51, P < 0.001). Age-related differences were larger in average distance and FD as compared with average time (F = 35.05, P < 0.001; Fig. 3). Independent of age, women traveled paths that on average covered a longer distance (β = −0.21, P = 0.003) and had a higher fractal dimensionality of the path (β = −0.20, P = 0.01) than men did (F = 8.22, P = 0.002), but no sex differences in travel times were observed: β = 0.05, P = 0.52. Age was unrelated to differences in joystick control (distance traveled to the visible platform; r = 0.09, P = 0.81) or prior video game experience (r = −0.15, P = 0.22); nor were there sex differences in either control measure (joystick control t = 0.52, P = 0.61; video game experience t = −0.42, P = 0.68). Further, joystick control (all P ≥ 0.07) and video game experience (all P ≥ 0.17) did not correlate with any trial navigation index or change in index across trials.

Figure 3.

The figures depict age differences in average time (A; r = 0.23, P = 0.01), distance (B; r = 0.51, P < 0.001), and FD (C; r = 0.52, P < 0.001) across the 6 learning trials. Error bars represent standard error of the mean.

Figure 3.

The figures depict age differences in average time (A; r = 0.23, P = 0.01), distance (B; r = 0.51, P < 0.001), and FD (C; r = 0.52, P < 0.001) across the 6 learning trials. Error bars represent standard error of the mean.

In the first HLM, we estimated change in time, distance, and FD across learning trials. Travel time (slope = −0.06, t(692) = −2.44, P = 0.02; Figure 4A) and distance (slope = −0.24, t(692) = −2.72, P = 0.01; Figure 4B) significantly decreased across the 6 learning trials. In contrast, improvement in FD across trials did not reach significance (slope = −0.001, t(692) = −1.70, P = 0.09; Figure 4C). However, individual differences in intercept and slope need to be examined as they can potentially explain age-related deficits in navigation.

Figure 4.

The figure shows the average change in time (A; P = 0.02), distance (B; P = 0.01), and FD (C; P = 0.09) across the 6 learning trials. Error bars represent standard error of the mean.

Figure 4.

The figure shows the average change in time (A; P = 0.02), distance (B; P = 0.01), and FD (C; P = 0.09) across the 6 learning trials. Error bars represent standard error of the mean.

For all indices, poor performance on the initial trial (i.e., the intercept) predicted greater improvement across trials (steeper slope). Participants who expeditiously found the platform on the first trial did not improve as much as those who had required more time (1.58, t(136) = 14.81, P < 0.001) or distance (8.62, t(136) = 30.11, P < 0.001) or had a higher FD (1.04, t(136) = 357.39, P < 0.001). For all indices, poor performance on the first trial was associated with advanced age (time = 0.01, distance = 0.02, FD = 0.0002; all t(136) >4.17, P < 0.001). Women required more time on trial 1 (−0.19, t(136) = −2.96, P = 0.004), but the sex difference in distance (−0.36, t(136) = −1.84, P = 0.07) and FD (−0.003, t(136) = −1.98, P = 0.05), while showing trends in the same direction, fell short of significance. Independent of the unique relationships with the intercept, neither age (all coefficients < 0.002, t(692) < 1.21, P ≥ 0.23) nor sex (all coefficients ≤ 0.01, t(692) ≤ 0.53, P ≥ 0.60) modified the rate of improvement across trials, as measured by any index.

Next, we examined the relationship between time, distance, and FD during navigation. A decrease in search path length across trials was associated with reduction in the search time (0.18, t(686) = 5.30, P < 0.001), independent of age (0.001, t(686) = 1.01, P = 0.31). Change in the fractal dimension of the search paths uniquely explained differences in navigation time across trials. Although on any trial FD and time were unrelated (−0.11 ≤ r ≤ 0.17, all P >0.05; see Supplementary Table 1), the decline in time of search was associated with the reduction in FD across trials (11.65, t(686) = 2.63, P = 0.01). That is, improvement in navigation across trials was in part characterized by decline in tortuosity of search paths. This relationship was independent of age (−0.012, t(686) = −1.30, P = 0.19) and sex (3.74, t(686) = 1.34, P = 0.18).

### Analyses of Brain and Behavior Associations

Smaller Hc volume (−0.000004, t(135) = −2.68, P = 0.01) was associated with larger FD on trial 1 (see Fig. 5) and the relationship with PHG volume exhibited a nonsignificant trend in the same direction (−0.000003, t(135) = −1.78, P = 0.08). Hc volume was associated with change in FD across trials (0.000001, t(691) = 2.58, P = 0.01), as was PHG volume (0.000001, t(691) = 1.99, P = 0.047). PHG volume was unrelated to differences in time (intercept = −0.0001, t(135) = −1.71, P = 0.09; slope = 0.00003, t(691) = 1.50, P = 0.14) and distance (intercept = −0.0004, t(135) = −1.72, P = 0.09; slope = 0.00008, t(691) = 1.22, P = 0.22). Greater travel time was associated with smaller Hc (intercept = −0.0001, t(135) = −2.22, P = 0.03; slope = 0.00002, t(691) = 1.97, P = 0.05) and Cb volumes (intercept = −0.001, t(135) = −2.36, P = 0.02; slope = 0.0002, t(691) = 2.07, P = 0.04), whereas distance was not (all t = −1.57–1.43, all P ≥ 0.12). FD was also unrelated to Cb volume (intercept = −0.0001, t(135) = −1.44, P = 0.15; slope = 0.00004, t(691) = 1.68, P = 0.09), and Cd (all t = −1.40–1.31, P ≥ 0.16), PFC (all t = −1.67–1.29, P ≥ 0.10), and VC (all t = −0.81–0.49, P ≥ 0.42) volumes were unrelated to any index of navigational performance.

Figure 5.

The scatter plot and regression of the relationship between fractal dimensionality on trial 1 and hippocampal volume. Gray circles mark data points that were winsorized.

Figure 5.

The scatter plot and regression of the relationship between fractal dimensionality on trial 1 and hippocampal volume. Gray circles mark data points that were winsorized.

## Discussion

To capture individual differences in search path complexity, we introduced a new measure of performance in a vMWM, fractal dimensionality, or FD. We demonstrated that FD contributes unique information about navigation in a virtual maze, including a specific pattern of age and sex differences and associations with regional brain volumes. The information provided by FD covers the aspects of virtual navigation that differ from those addressed by 2 traditional indices of performance: time of search and total distance traveled between the start and the goal.

Time of search declined in a log-linear fashion from the first trial to the last and the change in FD followed a similar pattern while not reaching statistical significance. The rate of search time shortening was associated with reduction in path tortuosity. Although advanced age was linked to higher average FD, progressively less meandering paths characterized faster learners, regardless of age and sex. The reasons for gradual simplification of the path are unclear. Spatial navigation is a complex cognitive activity that depends on allocentric and egocentric spatial cues, space-specific and general computational mechanisms, and spatial representations in real time and in memory (Wolbers and Hegarty 2010). Which of these cognitive processes are reflected in FD is an open question, but it is plausible that FD represents the formation and utilization of a cognitive map (see O'Keefe and Nadel 1978). Although the existence of a cognitive map is a core premise in theoretical models of spatial navigation, it has not been thoroughly studied and quantified in human navigation. FD of the navigational path provides a robust measure from which one may infer important properties of cognitive maps and elucidate some sources of individual differences in navigation efficiency.

Fractal dimensionality has been interpreted as an index of cognitive mapping in animal behaviors such as migration and foraging (Garcia et al. 2005; Gautestad and Mysterud 2010; Gautestad 2011). As a measure of path tortuosity, FD may reflect an individual's recognition and evaluation of landmark cues while seeking the goal. Reduction in FD across trials was associated with improvement in other indices of performance and may indicate the trial-by-trial improvement in the formation and utilization of a cognitive map. Indeed, higher FD (greater path tortuosity) was associated with smaller Hc volumes and showed a trend for a similar association with the volume of PHG, regions that have been implicated in age-related deficits in cognitive mapping (see Maguire et al. 1999 for a review). In fMRI studies of adult aging, hippocampal activation is correlated with spatial strategies employed in navigating virtual environment (Konishi et al. 2013), and the relevant activation in the medial temporal lobe decreases with age (Moffat et al. 2006; Konishi et al. 2013). Thus, allocentric navigation processes that typify the development of a cognitive map are associated with medial temporal lobe structures that show reliable age-related declines in structure and function. Allocentric navigation is hippocampus-dependent; however, the robust volumetric shrinkage that typifies aging (Raz et al. 2005) does not account for age-related deficits in vMWM as measured by distance (Moffat et al. 2007). The failure to find meaningful hippocampal differences in older adults is likely due to the lack of specificity of distance and time measures, the staple indices of navigation. In this study, we observed a selective association between FD path complexity and the volumes of 2 structures that have been consistently linked to navigation, cognitive maps, and place identification: Hc and PHG. Thus, FD may be a more sensitive and valid index of cognitive mapping than currently popular measures: time and distance.

Since its inception, the construct of cognitive mapping has been viewed as a reflection of multifaceted dynamic processes that unfold while navigating an environment (O'Keefe and Nadel 1978). Just as navigation is a complex skill that involves planning, timing, and memory, cognitive mapping relies on several functions (O'Keefe and Nadel 1978; see O'Keefe 1991 for a review). Age-related deficits in spatial memory (Moffat et al. 2001), mental rotation (De Simone et al. 2013), and path integration (Harris and Wolbers 2012) are associated with deficits in MWM navigation, and these functions are proposed as the major determinants of cognitive maps (O'Keefe and Nadel 1978; see O'Keefe 1991 for a review). Further, age-related deficits in cognitive mapping and navigation have been linked to smaller volume of the hippocampus, as well as in extra-hippocampal regions in the medial temporal lobe and striatum (see Moffat 2009 for a review). Here, we identify FD as a unique index of path complexity that unlike the traditional indices (time and distance of travel) selectively correlates with volumes of the Hc and PHG, which have been proposed as the neural correlates of cognitive mapping (O'Keefe and Nadel 1978; see O'Keefe 1991 and Maguire et al. 1999 for reviews). If reduction in path dimensionality reflects emergence of a more efficient mental map of the environment, FD may become a useful and sensitive index of quantifying that change.

In accord with the extant literature (Moffat et al. 2001; see Moffat 2009 for a review), we found that advanced age was associated with poorer navigation performance as assessed by traditional indices, time of search and the total distance traveled. Older participants needed more time to find the hidden platform and traveled longer distance to accomplish the task than their younger counterparts did. Similar results were observed with the new measure introduced here, fractal dimensionality. Older adults reduced search distance and FD across trials at a rate comparable with that of the younger adults. However, in spite of improvement, they did not reach an equivalent performance level as their younger counterparts by the last learning trial. Thus, advanced age is associated with slower navigation, but not impaired skill acquisition.

As is frequently documented in navigation studies (Sandstrom et al. 1998; Moffat et al. 2001; Driscoll et al. 2003; Nowak and Moffat 2011), men outperformed women in the initial trial of the vMWM. However, sex differences persisted in the process of navigation skill acquisition, but did not account for individual differences in learning slope (i.e., parallel slopes), or differences in distance-time or FD-time associations. Thus, as has been reported before (Moffat et al. 2001), it appears that women learned and improved at the same rate as men did, although the initial sex difference in the level of performance remained.

We observed several dissociations among structural brain correlates of various indices of navigational performance. Larger Hc was linked to less complex search paths, with a similar trend observed for PHG volumes. Whereas time required for finding the platform was negatively associated with the Cb and Hc volume, distance covered in the search was unrelated to any of the examined regional brain volumes. The age-related deficits that are indexed by increased distance and time of search have been hypothesized to reflect compromised cognitive processes that underpin navigation. However, these indices may be too coarse to tap into the underlying cognitive differences. Fractal dimensionality appears to reflect individual differences in search path geometry that are not captured by the total path length. Whereas FD represents path complexity, traditional measures of distance and time can be viewed as concise but coarse indices of navigation efficiency, that is, accomplishing the task in the shortest possible time and by presumably expending lesser energy in traveling along the shorter path. Path tortuosity informs about other aspects of navigation. Its differential association with medial temporal volumes suggests that it may reflect the aspects of mapping that are not accounted by global indices such as total time and total distance.

The pattern of individual differences varies between the 3 measures. The trial-by-trial changes in FD exhibited the same trend toward the monotonically declining trajectory that was observed for time and distance. However, across trials, FD approached 1, which corresponds to a straight line, or an ideal path to the goal. First-trial (presumably naïve) performance was characterized by search paths that had on average longer time and distance, and greater FD than performance on subsequent trials. Notably, even the first trial had a low FD approaching 1 (see Fig. 1A). The relatively low FD observed on the first trial suggests that even at the first attempt, participants already engaged in a nonrandom search toward the unknown goal location. When the goal location is unknown, participants may use a systematic search, while sacrificing efficient travel distance and time. After successfully finding the goal for the first time, participants who presumably have a memory of the location would apply different search strategies (O'Keefe 1991). Differences in strategy use partially account for age-related deficits in navigation (Moffat and Resnick 2002; Rodgers et al. 2012; Harris and Wolbers 2014; Wiener et al. 2013). Indeed, differences in strategy use with age may explain the differential recruitment of hippocampal and striatal regions (Bohbot et al. 2012; Konishi et al. 2013). Individual differences in strategy use may affect the development of a cognitive map (O'Keefe and Nadel 1978) and thus may also explain the reduction in FD across trials. Without reports of strategy use, we cannot presently test this hypothesis, and analysis of strategy use in future studies may shed light on its role in age-related differences in learning a navigation task.

## Funding

This study was supported by the National Institutes of Health grants R37 AG-011230 to N.R. and training grant T32 HS-013819 to the Institute of Gerontology, Wayne State University.

## Notes

Special thanks to Stewart Neufield (Institute of Gerontology, Wayne State University) for his assistance in evaluating the fractal dimension calculations. Conflict of Interest: None declared.

### Appendix

#### Behavior Model 1

Level 1 Model

$Y(time,distance,orFD)=γ0+γ1(trial)+ε.$

Level 2 Model

$γ0=β00+β01(age)+β02(sex)+r0,$

$γ1=β10+β11(age)+β12(sex)+β13(γ0).$

#### Behavior Model 2

Level 1 Model

$Y(time)=γ0+γ1(distance)+γ2(fractaldimension)+γ3(trial)+ε.$

Level 2 Model

$γ0=β00+β01(age)+β02(sex)+r0,$

$γ1=β10+β11(age)+β12(sex)+β13(γ0),$

$γ2=β20+β21(age)+β22(sex)+β13(γ0),$

$γ3=β30+β31(age)+β32(sex)+β13(γ0).$

#### Brain and Behavior Model 3

Level 1 Model

$Y(time,distance,orFD)=γ0+γ1(trial)+ε.$

Level 2 Model

$γ0=β00+β01(Age)+β02(sex)+β03(regionalbrainvolume)+r0,$

$γ1=β10+β11(age)+β12(sex)+β13(regionalbrainvolume).$

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