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Folia Neuropathologica
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vol. 58
Original paper

Fluctuations in neurofibrillary tangle density in Alzheimer’s disease revealed by Fourier (spectral) analysis

Richard Armstrong

Vision Sciences, Aston University, Birmingham, United Kingdom
Folia Neuropathol 2020; 58 (4): 299-306
Online publish date: 2021/01/11
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The presence of extracellular senile plaques (SP) and intracellular neurofibrillary tangles (NFT), which contain deposits of b-amyloid (Ab) [16] and phosphorylated microtubule associated protein (MAP) tau [17], respectively, are the ‘signature’ pathological features of AD [22,24]. In the cortex and hippocampus of patients with AD, regularly repeating patterns of Ab deposits [1] and NFT [5,19] have been observed in a direction parallel to the tissue boundary. These variations in density may reflect the degeneration of specific anatomical features [13,19,27], and studies of the spatial patterns of Ab deposits and NFT are potentially useful in elucidating the relationship between the pathological changes, the modular structure of the cortex [1,8], and the cerebral microvasculature [4]. In previous studies of the spatial patterns of brain lesions, the densities of NFT were recorded in ‘transects’ of contiguous sample fields aligned parallel with the tissue boundary [3]. The frequency distribution of the lesion densities in the sample fields was then compared with that predicted by the Poisson distribution; a variance/mean (V/M) ratio of the data less than or greater than unity indicating departures from randomness towards regularity or clustering, respectively [1,2]. Counts of lesions in adjacent sample fields may then be added together successively to provide data for larger field sizes, thus enabling cluster sizes to be estimated and the distribution of the clusters parallel to the tissue boundary to be determined [1].
Methods of spatial pattern analysis based on the Poisson distribution have a number of limitations [8]. First, the method can only be used to analyse counts of a lesion or ‘density data’ and not alternative measures of abundance such as ‘coverage’ or ‘load’ [3,9]. Second, a linear trend in the abundance of lesions parallel to the tissue boundary, or the presence of a single large cluster occupying a significant portion of a region, can make it difficult to detect smaller-scale clustering patterns [9]. Third, the method may not estimate cluster size accurately enough to be able to relate spatial patterns of a pathology to neuroanatomical features [1,2,5]. Fourth, in many cases of AD, NFT are present at high density, are not distributed in discrete clusters, and may exhibit a more continuous fluctuating pattern of abundance along cortical gyri parallel to the pia mater. Fifth, fluctuations in density may repeat at several different scales, and these more complex patterns are not well detected using the Poisson method [9].
Bruce et al. [12] were among the first authors to suggest that Fourier (spectral) analysis might be used to examine the laminar distribution of Ab deposits across the cortex from pia mater to white matter in AD [14]; a method subsequently used in the same disorder to analyse changes in Abdeposit density parallel to the pia mater [9]. Fourier analysis attempts to understand the pattern of fluctuations of a quantity in space or time (the ‘signal’) by breaking it down into simpler sine waves. The original signal comprises sine waves of different frequency corresponding to fluctuations repeating at different scales, the sine waves being easier to study and interpret than the original more complex signal. The method has been applied in many scientific fields including physics, mathematics, imaging, probability theory, and acoustics [30,31]. Hence, the objective of the present study was to determine the spatial fluctuations in NFT density in the cerebral cortex and hippocampus, measured parallel to the pia mater or alveus, respectively, and which constitute the ‘signal’. The analysis identifies the sine waves that comprise the frequency components of the NFT fluctuations, and these frequency components may then be interpreted in relation to the underlying neuroanatomical features of the cerebral cortex and hippocampus [9].

Material and methods


Six cases of sporadic AD (details in Table I), with particularly high densities of NFT, were obtained from the Brain Bank, Department of Neuropathology, Institute of Psychiatry, King’s College, London, UK. Informed consent was given for the removal of all tissue and the study followed the principles embodied in the 1964 Helsinki Declaration (as modified Edinburgh, 2000). Post-mortem delay was less than 20 hours in each case. The AD cases were clinically assessed, and all fulfilled the ‘National Institute of Neurological and Communicative Disorders and Stroke and Alzheimer’s Disease and Related Disorders Association’ (NINCDS/ADRDA) criteria for probable AD [34]. The histological diagnosis of AD was established by the presence of widespread neocortical SP consistent with the ‘Consortium to Establish a Registry of Alzheimer’s Disease’ (CERAD) criteria [24] and ‘National Institute on Aging (NIA)-Reagan Institute’ criteria [21]. The apolipoprotein E (APOE) genotype of the AD cases, an important risk factor for AD, was either 3/4 or 4/4 and the cases represented pathological stages V or VI of the Braak staging system [10,11]. In addition, all cases had NFT (B scores) of 3 as assessed using the National Institute on Aging-Alzheimer’s association (NIA-AA) guidelines [20,25].

Tissue preparation

Blocks were taken from: 1) the frontal lobe, to study the superior frontal gyrus (SFG) (BA 8,6); 2) the parietal lobe, to study the superior parietal lobe (SPL) (BA 7); 3) the occipital lobe (OC), to study the primary and secondary visual cortex (BA 17,18); and 4) the temporal cortex, to study the superior temporal gyrus (STG) (BA 38), lateral occipitotemporal gyrus (LOT) (BA 35), parahippocampal gyrus (PHG) (BA 28), sectors CA1-4 of the hippocampus, and dentate gyrus (DG). Tissue was fixed in 10% phosphate-buffered formal-saline and embedded in paraffin wax. 7-µm coronal sections were stained using the Gallyas silver-impregnation method [15], which reveals the cellular NFT particularly clearly (Fig. 1) [6]. The Gallyas stain has high sensitivity for detecting argyrophilic inclusions in cell bodies [23,33], PHF-1 and immunolabelling methods being less sensitive than the Gallyas method, and improved thioflavin-S stains [33]. In addition, the background staining associated with the Gallyas method enables recognition of anatomical landmarks and cytomorphology. The Gallyas stain is less useful, however, in studying the pathology in fine processes or in axon terminals.

Morphometric methods

In cortical gyri, with particularly high numbers of NFT, variation in density was studied from the crest to the base of a sulcus parallel to the pia mater, using two hundred 1000-µm sample fields arranged contiguously [3]. The sample fields were located in the upper cortex, approximating to layers II/III where AD pathology is most dense, although NFT pathology also occurs in lower cortical layers [5]. The short edge of the sample field was orientated parallel with the pia mater and aligned with guidelines drawn on the section. In the hippocampus, the sample fields were aligned initially with the alveus, and the pyramidal cell layer from sectors CA1 to CA3 were studied. Measurements were continued into sector CA4 using a guideline marked on the slide and which ceased approximately 400 µm from the dentate gyrus granule cell layer.

Data analysis

The data were analysed by single-spectrum (Fourier) analysis [30,31] using STATISTICA software (StatSoft Inc., 2300 East 14th St, Tulsa, Ok, 74104, USA). The analysis was applied to the NFT in each gyrus and hippocampus studied from each AD case with sufficient densities of NFT (a total sample of 36 regions). First, variations in the density of NFT along the gyri and CA region were examined for a linear trend. If such a trend was present, it might have obscured smaller-scale patterns, so the data were then ‘corrected’ to remove the effect of this linearity [9]. Second, the data from each region were tested to determine whether there were any statistically significant sinusoidal fluctuations present. If the fluctuations in density parallel to the tissue boundary were simply random ‘noise’, then the raw periodogram values (spectral densities) would follow an exponential distribution. Hence, the exponential distribution was fitted to the frequency distribution of periodogram values and goodness-of-fit tested using the Kolmogorov-Smirnov (KS) test [9]. Third, if the data deviated from an exponential distribution, significant recurring sinusoidal patterns were present, and the periodogram values were ‘smoothed’ to reduce the effect of measurement noise, the spectral density plot was then examined for significant peaks, which indicated the ‘periods’ of recurring fluctuations, i.e. the number of adjacent 200 mm diameter plots necessary to complete one full cycle.


An example of Fourier analysis applied to the NFT in the PHG of a single case (Case F) is shown in Figures 2-4. Figure 2 shows the variations in density of the NFT along the gyrus parallel to the pia mater revealing a complex fluctuating pattern and with an individual ‘spike’ in density at plot 65. Figure 3 shows the frequency distribution of the periodogram (spectrogram) values, a distribution that deviates significantly from the exponential distribution (KS = 0.26, p < 0.001), indicating the presence of sinusoidal fluctuations. Figure 4 is a plot of the spectral density at each period. A number of peaks or groups of peaks are evident: 1) a group of peaks at periods of 3-6 units representing a complete NFT density cycle with a mean at approximately 900 mm, 2) at a period of 15 units representing fluctuation of deposits on a larger scale with cycles recurring every 3000 mm, and 3) maximum spectral density at a period of 32 units representing an even larger scale of fluctuation repeating every 6400 mm.
A summary of the spectral analyses from all regions and cases is shown in Table II. Of the total of 36 analyses, 8/36 (22%) were corrected for a linear trend, 7/36 (19%) exhibited no statistically significant recurring sinusoidal pattern, and in 29/36 (81%) there was evidence of significant recurring fluctuations of NFT density of at least one frequency. Of the data exhibiting significant frequency components, 6/29 (21%) analyses exhibited a single frequency component (periods < 1000 mm in one analysis and > 1000 mm in five analyses), 11/29 (38%) exhibited two frequency components, and in five of these analyses, the first frequency component had a period of < 1000 mm and the second > 1000 mm. In 12/29 (41%) analyses, three or more frequency components were present. In 20/30 (66%) cortical regions, the periods were in the range 500-1000 mm, i.e. the size of the clusters of cells associated with the modular columns of cells that form the cortico-cortical pathways. The summary statistics for the distribution of the periods of the various frequency components in the AD cases are shown in Table III. In the cerebral cortex, the period of the first frequency component had a mean of 1519 mm (range 600-8000 mm), the second period a mean of 2767 mm (range 1000-12800 mm), and the third frequency component an overall mean period of 4078 mm (range 260-6400 mm). Periods of the first two frequency components were generally larger in the hippocampus than in the cerebral cortex.


In the majority of regions analysed, there were significant recurring patterns of NFT density in the upper cortical layers and in three out of the six hippocampus sections. These results are in general agreement with previous studies employing the Poisson distribution and suggest the presence of regularly repeating patterns of NFT formation in the cortex and the hippocampus in some cases in AD [1,5]. The commonest spatial pattern observed was a small-scale fluctuation in NFT density, a complete cycle repeating with a mean distance between 500 and 1000 mm, together with larger-scale patterns repeating at distances greater than 1000 mm. In some regions, particularly complex patterns were present with three or more frequency components, the largest-scale fluctuations having a mean period of at least 3733 mm. The anatomical structure of the cerebral cortex comprises replicated local neural circuits that represent ‘columns’ or ‘modules’ [26]. The diameter of the cortical modules is highly variable, depending on the region, and there are specific connections between ordered sets of columns [19,26]. First, there is a reciprocal projection between each cortical area and a dorsal nucleus of the thalamus. Second, there are inputs to the cortex from regulatory systems that have their origin in nuclei located in the basal forebrain and brain stem. As a result, the cortex is interlaced by clusters of fine noradrenergic fibres that repeat at intervals of 30-40 mm. Third, homologous neocortical areas, with the exception of the striate cortex and primary somatosensory cortex, are reciprocally linked in both hemispheres via the commissural projections; both the commissural and ipsilateral association fibres terminating in modules 200-500 mm in width, and which alternate in a regular sequence with modules that are relatively free of such connections. Fourth, within a hemisphere, different regions are connected by fibres of the short and long cortico-cortical projections [13,19], the cells of which are also clustered and occur in bands approximately 500-1000 mm in width.
A basic field size width of 200 mm cannot detect frequency components that repeat at intervals smaller than this size. Hence, it is not possible to determine whether any NFT were associated with the regulatory systems of the cortex. The smallest frequency component detected had a period consistently between 500 and 1000 mm. These dimensions suggest a spatial relationship between the NFT and the primary modules of the cortex and which form the cells of origin of the cortico-cortical projections, as postulated in previous studies [1,5,19,27,35]. The small number of gyri in which the period was significantly less than 500 mm could represent the association of NFT with the commissural projections. In the majority of gyri, more than one spatial frequency component was present, the most common spatial pattern representing fluctuations of NFT repeating every 500-1000 mm and at distances greater than 1000 mm. In the cerebral cortex, adjacent columns are often organised into larger functional units [26], and the larger-scale fluctuations could represent NFT formation in relation to these groupings of modules. In addition, NFT development could be a dynamic process with coalescence of smaller clusters of NFT to form much larger aggregations as the disease develops [2,35]. In approximately 22% of regions, there was a distinct linear trend in NFT density parallel to the tissue boundary, which may represent fluctuations on a significantly larger scale, perhaps reflecting the vulnerability of whole or adjacent groups of gyri.
Sinusoidal variations in NFT were also observed within the CA sectors of the hippocampus in three of the six AD cases. These fluctuations may also reflect underlying neuroanatomy and specifically the modular structure of the alvear pathway and perforant path [9]. Degeneration of the hippocampus and its isolation from the rest of the cerebral cortex as a consequence of these processes is likely to be an important factor in developing dementia in AD [13]. The association between cycles of NFT and the modular structure of the cortex has two implications for the pathogenesis of AD. First, the regular cyclic pattern of NFT appears to reflect their development in relation to neurons associated with specific neuroanatomical pathways. This hypothesis is supported by two observations: first, that NFT in AD are highly region specific, layer specific, and cell-type specific [10]; and second, that neurons affected by NFT are functionally related suggesting the spread of degeneration across normal synaptic boundaries [28]. In addition, loss of synaptic markers has been observed in AD, especially in layers III and V of the cerebral cortex, which are likely to contain the majority of cells of origin of the cortico-cortical projections [29]. Hence, the cyclic fluctuations in NFT appear to have their origin in the degeneration of specific anatomical pathways.
Of particular interest is whether the observed fluctuations in density of the NFT reflect the spread of pathogenic tau along neuroanatomical pathways. One of the first studies to suggest that the degeneration in AD could spread across synaptic connections was by Saper et al. [28]. More recent research suggests that pathogenic proteins, including tau, a-synuclein, the disease form of prion protein (PrPsc), and Ab may be secreted from cells, enter other cells, and seed small intracellular aggregates within these cells [18,32]. Hence, tau could exit cells via exocytosis or secretion and enter a new cell by endocytosis or by interactions with membrane lipids. Transfer may also occur via tunnelling nanotubes (TNT) that connect various neurons [32]. Hence, the spread of tau from cell to cell along neuro-anatomical connections in the cortex could be a significant factor in the observed fluctuations in NFT density detected by Fourier analysis. In conclusion, spatial analysis of NFT parallel to the pia mater or alveus in the cerebral cortex and hippocampus by Fourier (spectral) analysis reveals sinusoidal fluctuations in density. The fluctuations occur at different scales, the most common pattern being a small-scale sinusoidal fluctuation repeating every 500-1000 mm and a larger-scale pattern repeating at distances greater than 1000 mm. The data support the hypothesis that NFT formation is related to anatomical connectivity and most specifically to the degeneration of the callosal, cortico-cortical, and cortical-hippocampal projections. Hence, Fourier analysis, as suggested by Bruce et al. [12] and Armstrong and Cairns [9], may be of value in studying the detailed distribution of pathological lesions along or across cortical gyri in neurodegenerative disease and in transgenic models of disease, and further clarify the relative importance of NFT in the pathology of AD.


The author reports no conflict of interest.
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Copyright: © 2021 Mossakowski Medical Research Centre Polish Academy of Sciences and the Polish Association of Neuropathologists. This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0) License (http://creativecommons.org/licenses/by-nc-sa/4.0/), allowing third parties to copy and redistribute the material in any medium or format and to remix, transform, and build upon the material, provided the original work is properly cited and states its license.
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