BACKGROUND
three distinct processes underlie sleep regulation .
a homeostatic process, reflecting prior history of sleep and waking, plays a major role in sleep regulation.
additionally a circadian process, a clock-like mechanism independent of sleep and wake history, modulates sleep.
furthermore, an ultradian process occurs within sleep, reflected by the cyclic alternation of non-rapid-eye-movement  sleep and rapid-eye-movement  sleep.
the circadian and homeostatic processes were formalized in the two-process model of sleep regulation  <cit> .
according to this model, a homeostatic process  and a circadian process  interact to generate sleep-wake patterns.
slow-wave activity  serves as a marker for sleep homeostasis and thus for modeling of process s. swa shows a decline in the course of sleep that can be approximated by an exponential decrease across nrem sleep episodes  <cit> .
the level of swa in the first nrem sleep episode is dependent on the duration of prior waking and is best described with a saturating exponential function  <cit> .
thus, process s is characterized by an exponential decline during sleep towards a lower asymptote  and a saturating exponential increase towards an upper asymptote  during wakefulness.
process s was originally assumed to be a global process and the parameters of process s were mainly derived from central or fronto-central derivations .
bos et al.
<cit> , assuming that the eeg is the outcome of a stochastic process with similar or identical behavior over wide cortical areas, proposed that delta activity during sleep measured at different eeg derivations has a common source, but different gain factors and independent noise sources.
similarly, merica and fortune  <cit>  argued that sleep structure  results from brainstem mechanisms, whereas derivation specific aspects result from cortico-cortical interactions.
however, swa shows a frontal predominance and its percentagewise increase after sleep deprivation is most prominent in frontal areas  <cit> .
further evidence for regional and use dependent differences in the sleep eeg was obtained by experimental manipulation of specific brain regions during waking and the resulting local changes in the corresponding areas during subsequent sleep  <cit> .
such observations raise the question of whether the dynamics of process s show regional specificity.
our data base with  <dig> eeg derivations covering the scalp is well suited to address this question.
since sleep homeostasis is mainly reflected in the time constants, this paper focuses on the spatial distribution of the time constants.
furthermore, we examine the spatial distribution of the lower and upper asymptotes.
their absolute values reflect swa in an individual and exhibit age-dependent changes  <cit> .
in our previous publication  <cit>  we hypothesized that the distance between the normalized asymptotes  might reflect the capacity of the brain to generate slow waves.
thus, we expect that the distance between the asymptotes will also show regional differences.
we hypothesize that we will find regional differences in our measures of sleep regulation which reflect plastic brain processes occurring during wakefulness and sleep as articulated in the synaptic homeostasis hypothesis of tononi and cirelli  <cit> .
based on an elaborated model  <cit> , zavada et al.
<cit>  investigated regional aspects of swa regulation in recordings based on  <dig> eeg derivations.
in this model, the decline of process s was assumed to be proportional to the momentary level of swa.
zavada et al.
<cit>  reported a higher efficiency in swa dissipation in frontal derivations, a finding reflecting local aspects in sleep regulation.
however, they only examined the decline of the homeostatic process during baseline sleep.
we recently established a robust method for parameter estimation of process s on an individual basis and observed considerable inter-individual variation, particularly in the time constants  <cit> .
in this paper we investigate the spatial distribution of the parameters of process s within individuals by applying this new approach.
the data base consisted of eeg data of  <dig> derivations recorded during baseline sleep and during sleep following  <dig> h sleep deprivation.
because the spatial distribution of swa is characteristic for an individual  <cit>  and may reflect traits of functional anatomy, we also quantified inter-individual variation in the parameters of process s.
RESULTS
parameters derived from normalized averaged data
the homeostatic process s was modeled by a decreasing exponential function during sleep  and a saturating exponential function during waking .
the parameters of process s were first estimated for average data across subjects in each of the  <dig> derivations .
the corresponding topographical maps are illustrated in figure 1a.
in addition, the spatial distribution of swa for baseline sleep and the goodness of fit are displayed.
the time constants of the increase, τi, varied between  <dig>  and  <dig>  h .
the largest value, corresponding to the slowest buildup of process s, was found at the fronto-central derivation fc <dig>  whereas the fastest buildup was observed at f <dig> .
the time constants of the decline, τd, were in the range of  <dig>  to  <dig>  h  with the slowest decrease  of process s at the frontal-central derivation fc <dig>  and the fastest decrease at the occipital derivation o <dig> .
in general, a slower dynamic of process s was present in fronto-central areas compared to other regions.
range, mean value, standard deviation  and coefficient of variation  across the  <dig> derivations are shown.
τi, increasing time constant; τd, decreasing time constant; ua-la, distance between upper and lower asymptote.
τi, τd time constants of increase and decrease, and ua-la distance between upper and lower asymptote.
maxima  and minima  position of each parameter were localized in maps derived from parameter estimation in normalized average data over participants  and maps of mean parameters estimated with normalized individual data .
to assess topographical differences, estimated parameters in individuals at the corresponding location of the maxima and minima were compared by paired t-test .
electrode positions are indicated in figure  <dig>
the distance between the upper and lower asymptotes  showed a similar topographical pattern as swa in baseline and recovery sleep, i.e.
a frontal predominance, as well as the increase of swa from baseline to recovery sleep with the most prominent increase in frontal areas .
the goodness of fit was generally high  as expected for average data.
the best fit was observed in fronto-central areas whereas the worst fit occurred in temporal areas .
parameter distribution in individuals
next, the parameters of process s were estimated for each individual separately.
figure  <dig> shows the maps of the parameters estimated from normalized individual data.
although distinct individual spatial patterns were observed, the decline and buildup of process s were slowest in fronto-central areas across participants.
the fastest decreases occurred mainly in parieto-occipital areas whereas the fastest increases could not be assigned to a specific area.
the goodness of fit between simulations and empirical data was generally largest in fronto-central areas .
only one subject showed the largest goodness of fit in the centro-parietal region .
the location of the smallest goodness of fit varied between individuals and occurred primarily in temporal, parietal and occipital areas.
the variation of the time constants within and between individuals is depicted in the boxplots of figure  <dig>  non-overlapping notch areas between the different boxplots point to significant differences between individuals.
to assess the influence of the reference on the time constants, time constants estimated for derivation c <dig> against average reference and c3a <dig>  <cit>  are also indicated in figure  <dig> .
in most individuals the time constants of c <dig> determined with respect to these two references differed considerably but did not reach statistical significance.
maps averaged across individuals closely resembled those derived from average data  with value ranges of τi and ua-la slightly larger and values of τd slightly smaller than those for average data.
mean of goodness of fit showed a spatial distribution similar to average data but with lower values.
similarly, maps of the distance between upper and lower asymptotes  were comparable to the swa patterns .
however, the correspondence was less striking than those observed in the patterns of average data .
as maps of swa are characteristic for an individual  <cit>  and may reflect traits of the underlying functional anatomy, we compared maps of the distance between upper and lower asymptotes with swa maps to determine whether the same spatial pattern was exhibited in both.
to compare these maps, the manhattan distance  between normalized maps of ua-la and swa were examined.
the calculated mds were generally smaller within individuals than between individuals , except for subject  <dig>  suggesting to a larger similarity of the maps within individuals.
most importantly, the within-subjects distribution of mds differed significantly from the between-subjects distribution .
topographical differences in the dynamics of process s
topographical differences in the parameters of process s  were assessed by comparing the individual parameters at the location of maxima and minima determined in the maps of average data and in the mean maps of individually estimated parameters .
table  <dig> lists the corresponding electrode positions at which maxima and minima were observed.
parameter values at the position of maxima and minima differed significantly  pointing to regional differences in the dynamics of process s.
interestingly, maxima derived from average data and the mean of individual parameters were found at the same position  but not the minima of the time constants .
the minima in the distance between the asymptotes  were invariably observed at c <dig>
discussion
this was the first study analyzing topographical differences in the parameters of process s of the two-process model of sleep regulation  <cit> .
parameter estimation of process s was based on a recently established method  <cit>  which also allowed for parameter estimation in individuals.
also zavada et al.
<cit>  analyzed topographical aspects of sleep homeostasis.
however, they only focused on the decline of process s. for the first time, spatial aspects in the buildup of process s were investigated, which also allowed us to investigate differences between the asymptotes  and their relation to swa.
significant topographic differences in the parameters of process s were observed for both the decrease and the buildup of the homeostatic process s. distinct individual patterns in the spatial distribution of the parameters were observed .
these differences in the dynamics of process s contradict earlier views that postulated that swa measured at different derivations has a common source but different gain factors  <cit> .
merica and fortune  <cit>  investigated correlations between power in various frequency bands of a few derivations within single nrem sleep episodes.
they observed high correlations between derivations in all nrem sleep episodes.
however, such high correlations between derivations do not allow to conclude that the homeostatic process would show the same dynamics in all derivations.
homeostasis is mainly reflected in the time constants, i.e.
in the speed of the buildup and decay of the homeostatic process.
the faster the buildup , the quicker the upper asymptote is reached; this suggests that brain areas differ in their susceptibility to sleep deprivation.
the faster the decay , the quicker the lower asymptote is reached which indicates that brain regions differ in the speed of dissipation of sleep pressure.
swa showed a frontal predominance  and its increase after sleep deprivation was most pronounced in frontal areas.
one might have expected that the time constants of process s show an inverse spatial distribution to swa, with a fast  increase and decline of process s in frontal areas where swa is highest.
however, both the increase and decline of s were slowest in fronto-central areas.
the interpretation of these findings is difficult.
one hypothesis that remains to be tested could be that the differences in the dynamics of sleep homeostasis are related to the functional specialization of different cortical regions.
for example, frontal lobes are thought to be responsible for higher cognitive function.
as cognitive processing is essential for human functioning, it might be conceivable that these areas require a slower dynamic to minimize their vulnerability to challenges.
thus, it remains to be determined whether the speed of the deterioration of higher cognitive function during sleep deprivation is reflected in the time constants of the buildup  in frontal areas.
the synaptic homeostasis hypothesis as proposed by tononi and cirelli  <cit>  implies that the role of sleep is synaptic downscaling: this process reduces the synaptic strength that is high at the beginning of sleep due to plastic processes occurring during wakefulness.
similar to sleep homeostasis based on the eeg, the dynamics of synaptic homeostasis may show comparable differences in the corresponding spatial patterns.
however, it has been postulated that synaptic potentiation is also related to space requirements  <cit> .
in a recent paper we hypothesized , that the asymptotes might reflect the capacity of the brain to generate or produce slow waves.
in this paper we observed that the spatial distribution of the distance between upper and lower asymptote  closely resembled the spatial distribution of swa .
this finding supports our previous hypothesis.
then, if one assumes differences in available space for synapse formation during potentiation, it may be conceivable that the distance between the asymptotes reflects such an aspect resulting in brain region specific differences in the capacity to generate slow waves.
for both, the buildup and decline, distinct spatial patterns in the parameters of process s were observed which may be characteristic for an individual similar to the spatial distribution of power in different frequency bands  <cit>  and may reflect traits of the underlying functional anatomy.
because the topography of swa and the distance between the asymptotes showed very similar patterns and the topography of swa is an individual fingerprint  <cit> , we hypothesize that the distance between the asymptotes also represents an individual fingerprint.
whether this holds also for the spatial distribution of the time constants is an open question.
to assess whether time constants within an individual are stable over time, experiments with repeated sleep deprivation within an individual are required.
we analyzed whether the dynamics of sleep homeostasis, in particular the time constants, have trait-like aspects using existing data of a repeated sleep deprivation protocol  <cit> .
parameter estimation based on a single derivation indicated that the homeostatic process s is a trait  <cit> .
a number of studies have shown that the spatial distribution of swa changes with age.
for example, kurth et al.
<cit>  demonstrated that cortical maturation occurs in a region dependent manner and is reflected in swa, i.e.
the location of maximal swa undergoes a shift from posterior to anterior regions with increasing age.
furthermore, robillard et al.
<cit>  showed an age dependent reduction in the initial value of process s at sleep onset primarily in anterior regions of men which was more widespread in women.
the prominent increase in swa in frontal brain areas due to prolonged wakefulness also appears to be age-dependent  <cit> .
because we observed a close relationship between the topography of the distance between upper and lower asymptote and the spatial distribution of swa, we hypothesize that the distance between the asymptotes will also reflect developmental and age related alterations of brain morphology.
zavada et al.
<cit>  also investigated regional aspects of sleep homeostasis based on an extended model of ultradian swa regulation  <cit> .
in the extended model, the decay of process s is proportional to the momentary level of swa.
additionally, s has a permanent rise term.
the gain constant, gc, is used to determine the speed of the decline of s. thus, gc can be thought of as being approximately proportional to the inverse of the time constant of the decline .
to facilitate comparison with the maps of zavada et al.
<cit> , we plotted the decay rate rd as a grey scale map .
in the study of zavada et al.
<cit>  the topographical distribution of gc in different frequency bands was larger over frontal regions while our data revealed an occipital focus .
zavada et al.
<cit>  determined the decay based on baseline sleep and did not estimate the buildup of the homeostatic process based on data of sleep deprivation which may contribute to the disparate findings.
furthermore, zavada and colleagues kept the buildup fixed and equal in all individuals.
in our case, both, the buildup and the decay were estimated simultaneously based on baseline and recovery sleep after prolonged wakefulness.
given the large inter-individual differences observed in the dynamics of sleep homeostasis, the discrepancy between our data and the study of zavada et al.
<cit>  may simply be a consequence of inter-individual variation as different cohorts were investigated.
furthermore, zavada et al.
<cit>  estimated the decay rates in the 2- <dig> hz range along the anterio-posterior axis  by fitting exponential functions.
in figure  <dig>  we compared their decay rates with our own data.
the distribution of the decay rates along the anterio-posterior axis was similar, however, our decay rates were larger.
these differences in the absolute values of the decay rate may result from the different frequency ranges analyzed .
zavada et al.
<cit>  reported greatest incidence of simulation failures over occipital areas.
in our approach, the quality of fit  was best in fronto-central areas and worst in temporal areas.
this indicates that not all derivations can be fitted equally well.
however, parameter estimation converged in all individuals and at all derivations.
it is important to note that we limited time constants to a physiological meaningful range  <cit> .
zavada et al.
<cit>  proposed a distinction between a global process s related to the timing of sleep and waking which interacts with process c and a local homeostatic process related to swa regulation, which they called process z. based on their results of topographic differences in swa regulation, they concluded that process s and z are separate processes.
thus far there is no data to support a local cortical manifestation of process c. the interaction of a constant process c with a regional different process z would result in a regional different timing of sleep and waking.
therefore, such dissociation between a global process s and a local process z seems reasonable at first, if we accept a global timing of sleep and waking.
recent data, however, point to heterogeneity in the process of sleep onset in the range of several minutes in different cortical regions  <cit> .
this finding may question the assumption of a global timing of sleep and waking.
furthermore, it was recently shown, that most slow waves and their underlying active and inactive neuronal states occurred locally and not globally  <cit> .
it is difficult to reach a final conclusion regarding the significance of the observed topographic differences of the parameters of process s. however, these regional differences in the parameters may be related to plastic brain processes occurring during wakefulness and sleep  which may vary across brain regions.
if we accept that a global homeostatic process exists , the question arises how to measure and quantify the dynamics of such a global process, i.e.
how to estimate its parameters.
furthermore, time constants derived from c3a <dig> and c <dig> against average reference differed considerably .
thus, caution is necessary when comparing parameters derived from different derivations or from the same derivation with different references.
one limitation of our study is that only male subjects were investigated which limits generalization of our findings.
future directions would be to include women.
furthermore, the small sample size  may not allow us to capture the full range of inter-individual variation in the topography of sleep homeostasis.
nonetheless, the overall pattern of regional differences in sleep regulation was consistent across participants despite significant interindividual variability.
CONCLUSIONS
based on our recently established method, which allows for the estimation of the parameters of process s on an individual basis, we were able to investigate regional aspects of sleep homeostasis.
we found significant topographical differences in the dynamics of process s, for both, its buildup and decline.
in addition, individuals showed distinct spatial patterns in the parameters of process s.

 three distinct processes underlie sleep regulation. a homeostatic process, reflecting prior history of sleep and waking, plays a major role in sleep regulation. additionally a circadian process, a clock-like mechanism independent of sleep and wake history, modulates sleep. furthermore, an ultradian process occurs within sleep, reflected by the cyclic alternation of non-rapid-eye-movement sleep and rapid-eye-movement sleep. the circadian and homeostatic processes were formalized in the two-process model of sleep regulation. according to this model, a homeostatic process and a circadian process interact to generate sleep-wake patterns. slow-wave activity serves as a marker for sleep homeostasis and thus for modeling of process s. swa shows a decline in the course of sleep that can be approximated by an exponential decrease across nrem sleep episodes. the level of swa in the first nrem sleep episode is dependent on the duration of prior waking and is best described with a saturating exponential function. thus, process s is characterized by an exponential decline during sleep towards a lower asymptote and a saturating exponential increase towards an upper asymptote during wakefulness. process s was originally assumed to be a global process and the parameters of process s were mainly derived from central or fronto-central derivations. bos et al., assuming that the eeg is the outcome of a stochastic process with similar or identical behavior over wide cortical areas, proposed that delta activity during sleep measured at different eeg derivations has a common source, but different gain factors and independent noise sources. similarly, merica and fortune argued that sleep structure results from brainstem mechanisms, whereas derivation specific aspects result from cortico-cortical interactions. however, swa shows a frontal predominance and its percentagewise increase after sleep deprivation is most prominent in frontal areas. further evidence for regional and use dependent differences in the sleep eeg was obtained by experimental manipulation of specific brain regions during waking and the resulting local changes in the corresponding areas during subsequent sleep. such observations raise the question of whether the dynamics of process s show regional specificity. our data base with eeg derivations covering the scalp is well suited to address this question. since sleep homeostasis is mainly reflected in the time constants, this paper focuses on the spatial distribution of the time constants. furthermore, we examine the spatial distribution of the lower and upper asymptotes. their absolute values reflect swa in an individual and exhibit age-dependent changes. in our previous publication we hypothesized that the distance between the normalized asymptotes might reflect the capacity of the brain to generate slow waves. thus, we expect that the distance between the asymptotes will also show regional differences. we hypothesize that we will find regional differences in our measures of sleep regulation which reflect plastic brain processes occurring during wakefulness and sleep as articulated in the synaptic homeostasis hypothesis of tononi and cirelli. based on an elaborated model, zavada et al. investigated regional aspects of swa regulation in recordings based on eeg derivations. in this model, the decline of process s was assumed to be proportional to the momentary level of swa. zavada et al. reported a higher efficiency in swa dissipation in frontal derivations, a finding reflecting local aspects in sleep regulation. however, they only examined the decline of the homeostatic process during baseline sleep. we recently established a robust method for parameter estimation of process s on an individual basis and observed considerable inter-individual variation, particularly in the time constants. in this paper we investigate the spatial distribution of the parameters of process s within individuals by applying this new approach. the data base consisted of eeg data of derivations recorded during baseline sleep and during sleep following h sleep deprivation. because the spatial distribution of swa is characteristic for an individual and may reflect traits of functional anatomy, we also quantified inter-individual variation in the parameters of process s.  parameters derived from normalized averaged data the homeostatic process s was modeled by a decreasing exponential function during sleep and a saturating exponential function during waking. the parameters of process s were first estimated for average data across subjects in each of the derivations. the corresponding topographical maps are illustrated in figure 1a. in addition, the spatial distribution of swa for baseline sleep and the goodness of fit are displayed. the time constants of the increase, τi, varied between and h. the largest value, corresponding to the slowest buildup of process s, was found at the fronto-central derivation fc whereas the fastest buildup was observed at f. the time constants of the decline, τd, were in the range of to h with the slowest decrease of process s at the frontal-central derivation fc and the fastest decrease at the occipital derivation o. in general, a slower dynamic of process s was present in fronto-central areas compared to other regions. range, mean value, standard deviation and coefficient of variation across the derivations are shown. τi, increasing time constant; τd, decreasing time constant; ua-la, distance between upper and lower asymptote. τi, τd time constants of increase and decrease, and ua-la distance between upper and lower asymptote. maxima and minima position of each parameter were localized in maps derived from parameter estimation in normalized average data over participants and maps of mean parameters estimated with normalized individual data. to assess topographical differences, estimated parameters in individuals at the corresponding location of the maxima and minima were compared by paired t-test. electrode positions are indicated in figure the distance between the upper and lower asymptotes showed a similar topographical pattern as swa in baseline and recovery sleep, i.e. a frontal predominance, as well as the increase of swa from baseline to recovery sleep with the most prominent increase in frontal areas. the goodness of fit was generally high as expected for average data. the best fit was observed in fronto-central areas whereas the worst fit occurred in temporal areas. parameter distribution in individuals next, the parameters of process s were estimated for each individual separately. figure shows the maps of the parameters estimated from normalized individual data. although distinct individual spatial patterns were observed, the decline and buildup of process s were slowest in fronto-central areas across participants. the fastest decreases occurred mainly in parieto-occipital areas whereas the fastest increases could not be assigned to a specific area. the goodness of fit between simulations and empirical data was generally largest in fronto-central areas. only one subject showed the largest goodness of fit in the centro-parietal region. the location of the smallest goodness of fit varied between individuals and occurred primarily in temporal, parietal and occipital areas. the variation of the time constants within and between individuals is depicted in the boxplots of figure non-overlapping notch areas between the different boxplots point to significant differences between individuals. to assess the influence of the reference on the time constants, time constants estimated for derivation c against average reference and c3a are also indicated in figure. in most individuals the time constants of c determined with respect to these two references differed considerably but did not reach statistical significance. maps averaged across individuals closely resembled those derived from average data with value ranges of τi and ua-la slightly larger and values of τd slightly smaller than those for average data. mean of goodness of fit showed a spatial distribution similar to average data but with lower values. similarly, maps of the distance between upper and lower asymptotes were comparable to the swa patterns. however, the correspondence was less striking than those observed in the patterns of average data. as maps of swa are characteristic for an individual and may reflect traits of the underlying functional anatomy, we compared maps of the distance between upper and lower asymptotes with swa maps to determine whether the same spatial pattern was exhibited in both. to compare these maps, the manhattan distance between normalized maps of ua-la and swa were examined. the calculated mds were generally smaller within individuals than between individuals, except for subject suggesting to a larger similarity of the maps within individuals. most importantly, the within-subjects distribution of mds differed significantly from the between-subjects distribution. topographical differences in the dynamics of process s topographical differences in the parameters of process s were assessed by comparing the individual parameters at the location of maxima and minima determined in the maps of average data and in the mean maps of individually estimated parameters. table lists the corresponding electrode positions at which maxima and minima were observed. parameter values at the position of maxima and minima differed significantly pointing to regional differences in the dynamics of process s. interestingly, maxima derived from average data and the mean of individual parameters were found at the same position but not the minima of the time constants. the minima in the distance between the asymptotes were invariably observed at c discussion this was the first study analyzing topographical differences in the parameters of process s of the two-process model of sleep regulation. parameter estimation of process s was based on a recently established method which also allowed for parameter estimation in individuals. also zavada et al. analyzed topographical aspects of sleep homeostasis. however, they only focused on the decline of process s. for the first time, spatial aspects in the buildup of process s were investigated, which also allowed us to investigate differences between the asymptotes and their relation to swa. significant topographic differences in the parameters of process s were observed for both the decrease and the buildup of the homeostatic process s. distinct individual patterns in the spatial distribution of the parameters were observed. these differences in the dynamics of process s contradict earlier views that postulated that swa measured at different derivations has a common source but different gain factors. merica and fortune investigated correlations between power in various frequency bands of a few derivations within single nrem sleep episodes. they observed high correlations between derivations in all nrem sleep episodes. however, such high correlations between derivations do not allow to conclude that the homeostatic process would show the same dynamics in all derivations. homeostasis is mainly reflected in the time constants, i.e. in the speed of the buildup and decay of the homeostatic process. the faster the buildup, the quicker the upper asymptote is reached; this suggests that brain areas differ in their susceptibility to sleep deprivation. the faster the decay, the quicker the lower asymptote is reached which indicates that brain regions differ in the speed of dissipation of sleep pressure. swa showed a frontal predominance and its increase after sleep deprivation was most pronounced in frontal areas. one might have expected that the time constants of process s show an inverse spatial distribution to swa, with a fast increase and decline of process s in frontal areas where swa is highest. however, both the increase and decline of s were slowest in fronto-central areas. the interpretation of these findings is difficult. one hypothesis that remains to be tested could be that the differences in the dynamics of sleep homeostasis are related to the functional specialization of different cortical regions. for example, frontal lobes are thought to be responsible for higher cognitive function. as cognitive processing is essential for human functioning, it might be conceivable that these areas require a slower dynamic to minimize their vulnerability to challenges. thus, it remains to be determined whether the speed of the deterioration of higher cognitive function during sleep deprivation is reflected in the time constants of the buildup in frontal areas. the synaptic homeostasis hypothesis as proposed by tononi and cirelli implies that the role of sleep is synaptic downscaling: this process reduces the synaptic strength that is high at the beginning of sleep due to plastic processes occurring during wakefulness. similar to sleep homeostasis based on the eeg, the dynamics of synaptic homeostasis may show comparable differences in the corresponding spatial patterns. however, it has been postulated that synaptic potentiation is also related to space requirements. in a recent paper we hypothesized, that the asymptotes might reflect the capacity of the brain to generate or produce slow waves. in this paper we observed that the spatial distribution of the distance between upper and lower asymptote closely resembled the spatial distribution of swa. this finding supports our previous hypothesis. then, if one assumes differences in available space for synapse formation during potentiation, it may be conceivable that the distance between the asymptotes reflects such an aspect resulting in brain region specific differences in the capacity to generate slow waves. for both, the buildup and decline, distinct spatial patterns in the parameters of process s were observed which may be characteristic for an individual similar to the spatial distribution of power in different frequency bands and may reflect traits of the underlying functional anatomy. because the topography of swa and the distance between the asymptotes showed very similar patterns and the topography of swa is an individual fingerprint, we hypothesize that the distance between the asymptotes also represents an individual fingerprint. whether this holds also for the spatial distribution of the time constants is an open question. to assess whether time constants within an individual are stable over time, experiments with repeated sleep deprivation within an individual are required. we analyzed whether the dynamics of sleep homeostasis, in particular the time constants, have trait-like aspects using existing data of a repeated sleep deprivation protocol. parameter estimation based on a single derivation indicated that the homeostatic process s is a trait. a number of studies have shown that the spatial distribution of swa changes with age. for example, kurth et al. demonstrated that cortical maturation occurs in a region dependent manner and is reflected in swa, i.e. the location of maximal swa undergoes a shift from posterior to anterior regions with increasing age. furthermore, robillard et al. showed an age dependent reduction in the initial value of process s at sleep onset primarily in anterior regions of men which was more widespread in women. the prominent increase in swa in frontal brain areas due to prolonged wakefulness also appears to be age-dependent. because we observed a close relationship between the topography of the distance between upper and lower asymptote and the spatial distribution of swa, we hypothesize that the distance between the asymptotes will also reflect developmental and age related alterations of brain morphology. zavada et al. also investigated regional aspects of sleep homeostasis based on an extended model of ultradian swa regulation. in the extended model, the decay of process s is proportional to the momentary level of swa. additionally, s has a permanent rise term. the gain constant, gc, is used to determine the speed of the decline of s. thus, gc can be thought of as being approximately proportional to the inverse of the time constant of the decline. to facilitate comparison with the maps of zavada et al., we plotted the decay rate rd as a grey scale map. in the study of zavada et al. the topographical distribution of gc in different frequency bands was larger over frontal regions while our data revealed an occipital focus. zavada et al. determined the decay based on baseline sleep and did not estimate the buildup of the homeostatic process based on data of sleep deprivation which may contribute to the disparate findings. furthermore, zavada and colleagues kept the buildup fixed and equal in all individuals. in our case, both, the buildup and the decay were estimated simultaneously based on baseline and recovery sleep after prolonged wakefulness. given the large inter-individual differences observed in the dynamics of sleep homeostasis, the discrepancy between our data and the study of zavada et al. may simply be a consequence of inter-individual variation as different cohorts were investigated. furthermore, zavada et al. estimated the decay rates in the 2- hz range along the anterio-posterior axis by fitting exponential functions. in figure we compared their decay rates with our own data. the distribution of the decay rates along the anterio-posterior axis was similar, however, our decay rates were larger. these differences in the absolute values of the decay rate may result from the different frequency ranges analyzed. zavada et al. reported greatest incidence of simulation failures over occipital areas. in our approach, the quality of fit was best in fronto-central areas and worst in temporal areas. this indicates that not all derivations can be fitted equally well. however, parameter estimation converged in all individuals and at all derivations. it is important to note that we limited time constants to a physiological meaningful range. zavada et al. proposed a distinction between a global process s related to the timing of sleep and waking which interacts with process c and a local homeostatic process related to swa regulation, which they called process z. based on their results of topographic differences in swa regulation, they concluded that process s and z are separate processes. thus far there is no data to support a local cortical manifestation of process c. the interaction of a constant process c with a regional different process z would result in a regional different timing of sleep and waking. therefore, such dissociation between a global process s and a local process z seems reasonable at first, if we accept a global timing of sleep and waking. recent data, however, point to heterogeneity in the process of sleep onset in the range of several minutes in different cortical regions. this finding may question the assumption of a global timing of sleep and waking. furthermore, it was recently shown, that most slow waves and their underlying active and inactive neuronal states occurred locally and not globally. it is difficult to reach a final conclusion regarding the significance of the observed topographic differences of the parameters of process s. however, these regional differences in the parameters may be related to plastic brain processes occurring during wakefulness and sleep which may vary across brain regions. if we accept that a global homeostatic process exists, the question arises how to measure and quantify the dynamics of such a global process, i.e. how to estimate its parameters. furthermore, time constants derived from c3a and c against average reference differed considerably. thus, caution is necessary when comparing parameters derived from different derivations or from the same derivation with different references. one limitation of our study is that only male subjects were investigated which limits generalization of our findings. future directions would be to include women. furthermore, the small sample size may not allow us to capture the full range of inter-individual variation in the topography of sleep homeostasis. nonetheless, the overall pattern of regional differences in sleep regulation was consistent across participants despite significant interindividual variability.  based on our recently established method, which allows for the estimation of the parameters of process s on an individual basis, we were able to investigate regional aspects of sleep homeostasis. we found significant topographical differences in the dynamics of process s, for both, its buildup and decline. in addition, individuals showed distinct spatial patterns in the parameters of process s. 
swa shows a frontal predominance, and its increase after sleep deprivation is most pronounced in frontal areas. empirical mean swa per nrem sleep episode at episode midpoint served for parameter estimation at each derivation. furthermore, the spatial distribution of swa is characteristic for an individual and may reflect traits of functional anatomy. each individual showed distinct spatial patterns in the parameters of process s and the parameters differed significantly between individuals. time constants were restricted to a physiologically meaningful range. the question arises whether the dynamics of the homeostatic process s also show regional specificity. our data provide an additional indication of regional differences in sleep homeostasis and support the notion of local aspects of sleep regulation. for both, the buildup and decline of process s, significant topographic differences were observed: the decline and buildup of process s were slowest in fronto-central areas while the fastest dynamics were observed in parieto-occipital and frontal areas. sleep homeostasis refers to the increase of sleep pressure during waking and the decrease of sleep intensity during sleep. process s was modeled by a saturating exponential function during wakefulness and an exponential decline during sleep. the aim of the current study was to quantify inter-individual variation in the parameters of process s and investigate their spatial distribution. for the first time, topographical aspects of the buildup of process s were quantified. electroencephalography slow-wave activity is a marker of non-rapid eye movement sleep intensity and can be used to model sleep homeostasis.
 sleep homeostasis refers to the increase of sleep pressure during waking and the decrease of sleep intensity during sleep. electroencephalography slow-wave activity is a marker of non-rapid eye movement sleep intensity and can be used to model sleep homeostasis. swa shows a frontal predominance, and its increase after sleep deprivation is most pronounced in frontal areas. the question arises whether the dynamics of the homeostatic process s also show regional specificity. furthermore, the spatial distribution of swa is characteristic for an individual and may reflect traits of functional anatomy. the aim of the current study was to quantify inter-individual variation in the parameters of process s and investigate their spatial distribution. polysomnographic recordings obtained with eeg derivations of a baseline night of sleep and a recovery night of sleep after h of sustained wakefulness were analyzed. eight healthy young subjects participated in this study. process s was modeled by a saturating exponential function during wakefulness and an exponential decline during sleep. empirical mean swa per nrem sleep episode at episode midpoint served for parameter estimation at each derivation. time constants were restricted to a physiologically meaningful range.  for both, the buildup and decline of process s, significant topographic differences were observed: the decline and buildup of process s were slowest in fronto-central areas while the fastest dynamics were observed in parieto-occipital and frontal areas. each individual showed distinct spatial patterns in the parameters of process s and the parameters differed significantly between individuals.  for the first time, topographical aspects of the buildup of process s were quantified. our data provide an additional indication of regional differences in sleep homeostasis and support the notion of local aspects of sleep regulation. 