A 2-dimensional Geometry for Biological Time
AA 2-dimensional Geometry for Biological Time
Francis Bailly ∗ , Giuseppe Longo † , Mael Montevil ‡ October 23, 2018
Abstract
This paper proposes an abstract mathematical frame for describing some features of biologicaltime. The key point is that usual physical (linear) representation of time is insufficient, in ourview, for the understanding key phenomena of life, such as rhythms, both physical (circadian,seasonal . . . ) and properly biological (heart beating, respiration, metabolic . . . ). In particular, therole of biological rhythms do not seem to have any counterpart in mathematical formalization ofphysical clocks, which are based on frequencies along the usual (possibly thermodynamical, thusoriented) time. We then suggest a functional representation of biological time by a 2-dimensionalmanifold as a mathematical frame for accommodating autonomous biological rhythms. The “visual”representation of rhythms so obtained, in particular heart beatings, will provide, by a few examples,hints towards possible applications of our approach to the understanding of interspecific differencesor intraspecific pathologies. The 3-dimensional embedding space, needed for purely mathematicalreasons, allows to introduce a suitable extra-dimension for “representation time”, with a cognitivesignificance. Keywords: biological rhythms, allometry, circadian rhythms, heartbeats, rate variability.
Contents ∗ Physics, CNRS, Meudon † Informatique, CNRS – ENS and CREA, Paris, [email protected], ‡ Informatique, ENS and ED Frontières du vivant, Paris V, Paris Published in
Progress in Biophysics and Molecular Biology , 106(3):474 – 484, 2011. doi:10.1016/j.pbiomolbio.2011.02.001. a r X i v : . [ q - b i o . O T ] A p r More discussion on the general schema 1. 15 ( τ ) , its angles with the horizontal ϕ ( t ) and its gradients tan( ϕ ( t )) C e : its thread p e , its radius R i . . . . . . . . . 165.3 The circular helix C i on the cylinder and its thread p i . . . . . . . . . . . . . . . . . . 165.4 On the interpretation of the ordinate t (cid:48) . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Living phenomena displays rather characteristic and specific traits; among these, manifestations oftemporality and of its role are particularly remarkable: development, variegated biological rhythms,metabolic evolution, aging, . . . . This is why we believe that any attempt at conceptualizing lifephenomena — be it only partially — cannot avoid addressing such temporal aspects that are specificto it. In that which follows, we will examine this question from different angles in view of providing afirst attempt at synthesis.The intuitive “geometry of time” in physics was (and often still is) based, first, on the absoluteNewtonian straight time line. This was later enriched by the order structure of Cantor type realnumbers, an ordered set of points, topologically complete (dense and without gaps). Thermodynamicsand the theories of irreversible dynamics (phase transitions, bifurcations, passing into chaos, . . . )have imposed an “arrow” upon classical time, by adding an orientation to the topological and metricstructure. But it is with relativity and quantum physics that the theorization of time has led to ratheraudacious reflections. In the first case, to give only one example from a very rich debate which goes sofar as to introduce a circular time (proposed by Gödel as a possible solution to Einstein’s equations)to Minkowski space: by means of its famous causality cone, this space explains, within the frameworkof a unified geometry of space-time, the structure of any possible correlation between physical objects,in special relativity.In quantum physics the situation is maybe even more complex or, in any event, less stable. Wego from essentially classical frameworks to a sometimes two-dimensional time (in accordance with thestructure of the field of complex numbers with regard to which Hilbert spaces are defined, the theoreticalloci of quantum description), up to the audacity of Feynman’s temporal “zigzags” [FG67]. This latterapproach is a very interesting example of intelligibility by means of a “geometric” restructuring of time:the creation of antimatter would cause within the
CP T symmetry (charge, parity, time) a symmetrybreaking in terms of charge, while leaving parity unchanged. Global symmetry is then achieved bylocally inverting the arrow of time. Another approach, with similar motivations, is that of the fractalgeometry of space-time, specific to the “scale relativity” proposed by [Not93]: in it, time is reorganizedupon a “broken” line (a fractal), which is continuous but non-derivable. Further interesting reflections,along similar lines, may be found in [LMNN98].Physics however will remain but a methodological reference for our work, because the analysisof the physical singularity of living phenomena [BL06, BL11] requires a significant enrichment of theconceptual and mathematical spaces by which we make inert matter intelligible. One of the newfeatures which we introduce consists in the usage that we will make of the “compactification” of atemporal straight line: in short, we will try to mathematically understand rhythms and biological cyclesby means of the addition of “fibers” (a precise mathematical notion, introduced summarily below) whichare orthogonal to a physical time that remains a one-dimensional straight line . From our standpoint,a living being is a true “organizer” of time; by its autonomy and action, it confers it a more complexstructure than the algebraic order of real numbers, but also more than any organization one couldpropose for the time of inert matter. In short, the time of a living organism, by its specific rhythms,intimately articulates itself with that of physics all the while preserving its autonomy. We wouldtherefore like to contribute to making the morphological complexity of biological time intelligible, bypresenting a possible geometry of its structure, as a two dimensional manifold.2he first paragraph will introduce the theme of biological rhythms. One consequence of our ap-proach is the possibility of mathematically giving what we hope to be more precise and relevantmeaning to notions that are usually treated in a rather informal fashion and unrelated between oneanother, such as those of representation time, physical time vs. biological rhythms, . . . and this withina rigorous mathematical frame.
Let’s recall that physics, in its history, was constituted according to major dimensional constants(gravitation, the speed of light, Plank’s constant — with dimensions, respectively: acceleration, speed,action). What is striking, in biology, is the presence of a few major invariants with no dimension , thosethat are specified in the rhythms of which we will speak below. The mathematization of physics con-centrated on invariants, among which the above constants, but also those of “objective determinations”,which we address in length in [BL06, BL11]. We suggest here to start with these rare invariants, theseconstants and rhythms which are to be found in biology, because, beyond the physico-chemical, the structural stability of living phenomena is not “invariant”, physically speaking: it is profoundly imbuedwith variability .Observe also that in physics, time is mostly described as a parameter of the state functions de-scribing a system. The phenomena encountered in biology, however, seem to trigger the need of othertheoretical strategies and this at many different temporal levels of organization (physiology, ontogene-sis, phylogenesis, . . . ). We will provide a geometrical scheme of biological time that stresses the crucialrole of time in life and allow to understand some of the above features mainly through the use of twotheoretical concepts.The first one, which we will discuss in depth latter, is the ubiquity of rhythms in biological temporalorganization. There is indeed few features that are ubiquitous in biology but the iteration of similarprocesses seems to be one of them. We will however make a clear distinction between two type ofcyclicity encountered in living systems.The second concept is a way to understand the constitution and maintenance of biological orga-nization, both in phylogenesis and embryogenesis, that we formalized by the notion of anti-entropyin [BL09]. That approach allows the addition of a new theoretical aspect of time irreversibility inbiological systems, that completes and adds up to the thermodynamical irreversibility driven by thenotion of entropy. At the level both of evolution and embryogenesis, this irreversibility manifests itselfby the increase of complexity of the organism (number of cells, number of cell types, cell networks— neural typically, geometrical complexity of the organs, constitution of interacting yet differentiatedlevels of organization, . . . ).Methodologically, by a duality with physics, in [BL09] time is understood as an operator (like energyin Quantum Physics), not as a parameter. This makes time a fundamental observable of biology (likeenergy in physics) and it gives meaning to its key role in “biological organization”, since rhythmsorganize life.
We will introduce a second dimension of time, associated to the endogenous internal rhythms of or-ganisms, a dimension of time which we will represent as compacified ( S topology ).We denote this compacified time as θ , which we can represent as a sort of “circle” with a “radius” R i (where R i is the proper biological time): this circle expresses the temporal circularity, the iterative The circle is the compactification of the real number straight line, by the addition of a point and its folding.
We will distinguish two types of rhythms associated with biological organization, each referring to adistinct temporal dimension (below, we will note them as t and θ , respectively): (Ext) “external” rhythms, directed by phenomena that are exterior to the organism, with a physicalor physico-chemical origin and which physically impose themselves upon the organism. So theserhythms are the same for many species, independently of their size. They express themselvesin terms of physical, hence dimensional, periods or frequencies (s, Hz) and the invariants aredimensional; they are described with regard to the dimension of physical time (in exp( ıωt ) ).Examples: seasonal rhythms, the 24 hours-cycle and all their harmonics and sub-harmonics, therhythms of chemical reactions which oscillate at a given temperature, etc. (Int) “internal” rhythms, of an endogenous origin, specific to physiological functions of the organism,depend on purely biological functional specifications. These rhythms are characterized by periodswhich scale as the power / of the organism’s mass and, when related to the life span of theorganism which scales in the same way, they are expressed as pure numbers (they have no physicaldimensionality). Invariants are therefore numeric. We propose to describe them with regard toa new compacified “temporal” dimension θ , with a non-null radius, the numeric values thencorresponding to a “number of turns”, independently of the effective physical temporal extension(we have mentioned some examples: heartbeats, breathings, cerebral frequencies, etc.).We will now, even if we must be somewhat repetitive, describe further how these rhythms takeplace in biological organization, which is precisely what we would like to provide account for: • The external rhythms (Ext) are to be identified with physical time (typically measured by aclock) determined universally — their temporal features does not depend of the biological systemwe consider. Key examples include circadian, circannual or tidal cycles. The effects or therelevancy of these cycles depend of course on the organism that we consider (with possible sexualdimorphism). For example, diurnal and nocturnal animals are in phase opposition, whereas tidesare mainly relevant for marine organisms, and especially in the foreshore. Whatever organismwe consider, the period and the phase of these rhythms are the same as they are dependenton external physical events. In order to be a little more precise, this rhythms are generallyassociated with a double process: the physical process, outside the living system (and which canbe very precisely predicted) and its “shadow” inside the system which is kept synchronized byso-called “Zeitgeber” (light for circadian cycle for example). This distinction leads in particularto a specific inertia, encountered for example in the “jet lags” phenomenon.Simple chemical oscillations inside an organism will fall in this category too, since their period isdetermined by physical principles, however their phase depends on a specific organism (a specifictrajectory) since it is the organism which constructs this chemical system.As a result, this kind of rhythms, and their subharmonics, can be considered in the usual physicalway and represented by terms like e ıωt . • The second kind of rhythms, the endogenous biological cycles in (Int), do not depend directly onexternal physical rhythms. They could be called autonomous or eigen rhythms and scale withthe size of the organism (frequencies brought to a power − / of the mass, periods brought to apower / ), which is not the case with constraining external rhythms which impose themselvesupon all (circadian rhythms, for example). Such rhythms are encountered when we consider theheart rate, the respiratory rate, the mean life span, . . . , see [SGW +
04] or [LCI81] for example.4his rhythms are naturally associated with the number of their iterations (they can be seen asdual variables), and these numbers provide a natural way of speaking of the age of a biologicalsystem, yet different of the time measured by a clock. The distinction between replicative andchronological aging for yeasts, is a clear example of this situation, see [FPM + properties, a clear and impressiveexample of this is the mean number of heartbeat (or respiration) during life which is almostinvariant among mammals.In summary, endogenous biological rhythms: • are determined by pure numbers (number of breathings or heart beats over a lifetime, for example)and not, in general, by dimensional magnitudes as is the case in physics (seconds, Hertz, . . . ); • depend on the adult mass of the organism that we consider, by following the allometric law τ i ∝ W / f (for heterotherms, the temperature is involved too); • in our approach, they are analyzed and put into relation to each other by adding an additionalcompacified “temporal” dimension (an angle, actually, like in a clock), in contrast to the usualphysical dimension of time, a line, non-compacified and endowed with dimensionality.Since these endogenous rhythms co-exist with physical time, we consider a temporality of a topo-logical dimension equal to formed by the direct product of the non-compacified part, the real straightline of the variable t (the physical time parameter) and, as a fiber upon the latter, the compacifiedpart, a circle S , of which the variable is θ . Since we consider a two-dimensional time, with a seconddimension associated with specific biological invariants, our approach is very different of the usualapproach of biological time in terms of dynamical systems, which allows to tackle different kind ofquestions, like synchronization or stability (see for example the noteworthy book of [Win01]), but donot deal with these invariants.The idea of using supplementary compacified dimensions in theoretical physics has been introducedby Kaluza and Klein [OW97], and is still widely used in unification theories (string, superstring, M-theory, . . . ). There are of course major differences between these uses of compacified dimensions andours. First they concern mostly the addition of space- dimensions; second these dimensions are notobservable in physics, whereas they are clearly observable in biology. In our approach, the projectionof this second dimension on physical time leads to quantities that have the dimension of a time; theirmean follows the allometric law, as such they are parametrized by a mass (or, equivalently, by anenergy; here one may see again the dual role of energy vs. time as parameter vs. operator, the dualityw. r. to Quantum Physics we mentioned and extensively used in [BL09]).We insist that the endogenous rhythmicities and cyclicities are not physical temporal rhythms orcycles as such, as they are iterations of which the total number is set independently from the empirical(temporally physical) life span. As we said, they are pure numbers, a few rare constants (invariants)in biology. Our aim is that of a geometric organization of biological time which, by the generativityspecific to mathematical structures, would also enable us to derive meaning and to mathematicallycorrelate diverse notions. The text itself constitutes the commentary and the specification of thefollowing schemata, which are meant to “visualize” the two-dimensionality which we propose for thetime specific to living phenomena. There is still some variability, but this variability appears “naked” when considering this numbers, whereas the massand temperature effects come first when considering dimensional quantities. Mathematical description
We first consider both external and internal rhythms; later, we will mainly focus on internal rhythms oforganisms (we can take as a paradigmatic example the heart rate of mammals). We begin by providinga qualitative draft of our scheme to show its geometrical structure in figure 1, then we will quantifyits parameters and explain more precisely their meaning.
Following the aforementioned ideas, biological time is a (curved) surface: thus, it will be describedin 3-dimensions (the embedding space). Note that, if we were considering only biological rhythms,our 2-dimensional manifold would be a cylinder: the (oriented) line of physical time times the extracompacified dimension. The situation is more complicated, in view of the further, physical rhythms wewant to take into account. They do not require an extra dimension, but they “bend” the cylinder, byimposing global (external) rhythms. Thus, a proper biological rhythm is represented by what we maycall a “second order helix”, that is, a helix that is obtained (is winding) over a cylinder, C i , which, inturn, is winding around a bigger cylinder, C e , of which the axis is the line ( τ ) . As basic reference, wechoose orthogonal Cartesian coordinates. Physical time, which is oriented by thermodynamic principlesof irreversibility and is measured by a clock as in classical or relativistic physic, will be the first axis ( t ) of our reference system and will enable the characterization of instants and the measurement ofdurations. The second axis, ( t (cid:48) ) , will be associated with the proper irreversibility of biological time(for example the irreversibility of embryogenesis or, just, of “living”, see 5). As such, it will representthe biological age , or the internal irreversible clock of the organism we consider. The ( t ) and ( t (cid:48) ) axisare oriented in the usual way ( ( t ) towards the right and ( t (cid:48) ) pointing upwards). The third axis, ( z ) ,(see 1) is generated by the mathematical need of a 3-dimensional embedding space; yet, we claim thatit has a biological meaning that will become clear later, in section 3.2.Figure 1: Qualitative illustration of our geometric scheme, as a 2-dimensional manifold.
In red, theglobal age of the organism, in blue its modulation by the physical rhythm. Here the surface is suggestedby varying values of θ . 6e consider the surface of the cylinder C i as parametrized by t (the physical time) and θ ∈ [0 , π ] (the compacified time).Let’s then take a further step by gradually making explicit the functional dependencies. • The average progression with respect to ( t (cid:48) ) will be represented by a function τ (cid:16) t − t b τ i (cid:17) . τ is agrowing function due to the irreversibility of biological time, and has a decreasing derivative dueto the decrease of activity during development and aging. t b is the physical time of a biologicalevent of reference (time of fecundation for example). τ i is a characteristic time of the biologicalactivity of the adult: for example, the mean “beat to beat” interval under standardized conditions(other reference systems can be chosen such as the mean time taken to attain % of adult mass,life expectancy, respiratory interval, . . . ). This value represents, as a function of physical time,the age of the system inasmuch this age is biologically relevant (see figure 2a: the graph of τ lieson the ( t × t (cid:48) ) plane). Set then: −→ F τ i ( t, θ ) = tτ (cid:16) t − t b τ i (cid:17) (1) • We next consider a physical (external) rhythm of period τ e (its pulsation is then ω e = πτ e ) thataffects the activity rate of the organism — the circadian rhythm, for example, leads to τ e = 24 hours. This produces a winding spiral or helix, C e (see figure 2b: here we need the third dimension ( z ) for the embedding space of our manifold). In the definition of −→ G τ i ( t, θ ) , R e represents theimpact of this physical rhythm on biological activity: −→ G τ i ( t, θ ) = −→ F τ i ( t, θ ) + R e ω e τ i cos ( ω e t ) R e ω e τ i sin ( ω e t ) (2)The term ω e τ i is proportional to the number of iterations of the compacified time during oneperiod of the physical rhythm, as such it can be considered as the temporal weight of this rhythmfor an organism (mean number of heartbeat during a day, for example), it allows to understandthat a year is more important for a mouse and for an elephant. As a consequence the radius of C e is proportional to both the impact R e of this rhythm on biological activity, and on the weightof this rhythm in terms of number of iteration of the endogenous rhythm considered during oneperiod of the physical rhythm. • We can finally add a biological (internal) rhythm, which depends on an increasing function s τ i ( t ) (see figure 3). s τ i ( t ) has a proper biological meaning: for example, if we impose s τ i ( t b ) = 0 , with t = t b when the heart begins to beat , s τ i ( t ) is the number of heartbeats of the organism attime t , and thus the mean maximum of s , obtained when death occurs, does not depend on theorganism we consider (among mammals, typically). Set then, for −→ G τ i ( t, θ ) as in equation 2: −→ T τ i ( t, θ ) = −→ G τ i ( t, θ ) + R i cos (2 πs τ i ( t ) + θ ) R i sin (2 πs τ i ( t ) + θ ) (3) Let’s remark that, unlike in physics — classical, relativistic or quantum— biological time has an origin, whateverlevel of organization we consider. As a result there is no time-symmetry for translations, a fundamental property, in(relativistic) physics for the constitution of invariants, e.g. energy conservation. a) (b) Figure 2:
Qualitative Illustration of the first components of our model.
Left , the function τ (cid:16) t − t b τ i (cid:17) ,which represents the global age of an organism: this age increases at a greater pace during developmentand slows down progressively, see section 5. In orange a small mammal (a mouse for example) and inred a bigger one (an elephant). The life span of the first is shorter than one of the second. Right , inblue (and yellow), a physical rhythm has been added (this rhythm is very slow for illustrative purposes).Notice that this physical rhythm has the same period for both animals, but one of its iteration has agreater weight for the smaller animal.
Now, one of the simplest way to define more precisely s is to use −→ G τ i and more precisely the length ofthe curve defined by −→ G τ i . We obtain then for the instantaneous pulsation, where τ (cid:48) is the derivativeof τ (thus ddt τ (cid:16) t − t b τ i (cid:17) = τ i τ (cid:48) (cid:16) t − t b τ i (cid:17) ) and the other components are the derivative of the remainingcoordinates in equation 3: ds τ i ( t ) dt = (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) α × + τ (cid:48) (cid:16) t − t b τ i (cid:17) τ i − R e τ i sin ( ω e t ) + (cid:18) R e τ i (cid:19) cos ( ω e t ) (4)The term α is here for (physical) dimensionality reasons: since our metric has the dimension ofa frequency, and dtdt = 1 , the derivative of the first component of the vector in equation 2, has nodimension, then we need to introduce this coefficient whose dimension is a frequency .When α = 0 we can simplify 4 to: ds τ i ( t ) dt = (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) τ (cid:48) (cid:16) t − t b τ i (cid:17) τ i + (cid:18) R e τ i (cid:19) − R e τ (cid:48) (cid:16) t − t b τ i (cid:17) τ i sin ( ω e t ) (5)Now, if we consider hibernating animals, or frozen organisms, we have situations where the physicaltime flows normally but where the biological time almost stops or even totally stops. For α (cid:54) = 0 , evenin the frozen case, biological time would flow with ddt s τ i ( t ) ≥ α . It seems then natural to suggest This kind of reasoning is commonplace in physics. α = 0 . Moreover, for α = 0 , we go back to allometric relations, since, in this case, ddt s τ i ( t ) is proportional to τ i . Now, τ i is proportional to W / f , by allometry, and, thus, ddt s τ i ( t ) , which is afrequency, to W − / f , as it should be.Another way to express this is to say that physical time per se does not make biological organizationget older: it is only when there is a biological activity (which in return is of course always associatedwith physical time) that aging appears.We can now even give a meaning to the third, ( z ) , axis: since τ (cid:16) t − t b τ i (cid:17) is on the ( t × t (cid:48) ) plane, apositive ( z ) corresponds to a positive sin ( ω e t ) , by equation 3, and it is associated with a slowdownof biological activity (sleep, for example), whereas the negative values are associated to a faster pace(wake for example).As a fundamental feature of the model that we will analyze next, we assume that the speed ofrotation with respect to the compacified time is constant, which leads to a radius R i = Cst. (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ −→ T τ i ( t, θ ) ∂θ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = R i ( t, θ ) = Cst (6)This assumption “geometrizes” time even further: acceleration and slow-down will be seen ascontraction and enlargement of a cylinder in §4.4.2. In that section, as an application, we will developa geometrical analysis of biological rate variability, and, as an example, we will consider heart rate. Notethat this radius R i is the dimension accommodating the biological rhythms, thus it is not a physicaldimension (it is a pure number). Our assumption is consistent with the idea that each iteration alongthe compacified time contributes equally to aging. In this section we will explore the various biological aspects our approach allows to put together, mainlyon the questions of interspecific and intraspecific allometry and on (heart) rate variability.
Since ddt s τ i ( t ) provides the frequency of the biological rhythm, it is interesting to look for a simpleanalytical expression of the period associated. To do so, we perform a Taylor development (under thehypothesis τ (cid:48) (cid:16) t − t b τ i (cid:17) (cid:29) R e ) of the inverse of equation 5, and as a result we obtain an approximation ofthe physical time associated with an iteration of the compacified time (the time between two heartbeatsfor example): ds τi ( t ) dt (cid:39) τ i τ (cid:48) (cid:16) t − t b τ i (cid:17) + R e τ (cid:48) (cid:16) t − t b τ i (cid:17) sin ( ω e t ) (7)We can observe several things here. First, for adults (i.e.: τ (cid:48) (cid:16) t − t b τ i (cid:17) (cid:39) Cst and it does not depend onthe size of the organism we consider) the result has the form τ i ( a + b sin ( ω e t )) . As a consequence, whenwe take different species, there is no variation of the ratio ( τ i bτ i a ) between the continuous and the periodcomponents of the biological rates. Alternatively the ratio between the rates of the biological rhythmduring the slow period of the physical rhythm (sleep for example) and during the fast period (wake)does not depend of the species either. This result holds experimentally (see for example [SGW + τ (cid:48) is not constant, mainly during development), and the variation of the coefficient of9he rhythmic component R e τ (cid:48) (cid:16) t − t b τ i (cid:17) − is far greater than that of the steady (continuous) component τ (cid:48) (cid:16) t − t b τ i (cid:17) − . This mathematical deduction agrees with experimental results, since [MMW + t . while the sinusoidal part (associated withthe circadian rhythm ) varies like t . for humans (between 2 months and 15 years). (a) Cofluent case (b) Minimally cofluent case (c) Non-cofluent case(d) Cofluent case (e) Minimally cofluent case (f) Non-cofluent case Figure 3:
Illustration of the three scenari.
Top : the scheme −→ T τ i ( t, θ ) and bottom its time derivative ∂ −→ T τi ( t,θ ) ∂t . From left to right : Cofluent case, minimally cofluent case and non-cofluent case. Sincethe radius of the compacified time is proportional to its physical rate when looking at ∂ −→ T τi ( t,θ ) ∂t (see§4.4.1), the bottom pictures allows to see when the slowest rate occurs (i.e.: when the radius is thesmallest, blue arrow. Here respectively: for adults in figure 3d and 3e and for infants in 3f).We can now look more precisely at the second axis, ( t (cid:48) ) , of our reference system. Since this aspectof biological time is irreversible and flows in the same direction than physical time ( τ ( t ) is an increasingfunction of t ), −→ G τ i in equation 2 should increase with respect to this direction. When this conditionis met, we will say that these times are “ cofluent ”. This can be easily mathematized by looking at thepartial derivative of the ( t (cid:48) ) component of −→ G τ i (obtained with the dot product by the unitary vector −→ e t (cid:48) ) with respect to t : ∂ −→ G τ i ( t, θ ) ∂t . −→ e t (cid:48) = 1 τ i f (cid:48) (cid:18) t − t b τ i (cid:19) − R e τ i sin ( ω e t ) (8)We obtain then three different scenari, assuming that τ (cid:48) (cid:16) t − t b τ i (cid:17) tends to be a constant for adults(and seniors), written τ (cid:48) (cid:16) t ∞ τ i (cid:17) . We then use equation 5 to derive their observable consequences:10 (cid:48) (cid:16) t ∞ τ i (cid:17) > R e . In this case, biological age and the physical clock are cofluent, and the minimum rateis achieved during adult sleep (figure 3a and 3d). τ (cid:48) (cid:16) t ∞ τ i (cid:17) (cid:39) R e . In this case, they are minimally cofluent, the derivative tends to zero (during nightor winter) when the organism grows older, that is the rate of the biological rhythm tends to during the (physical) time of little biological activity. It seems to be particularly relevant forhibernation (figure 3b and 3e). . . . τ (cid:48) (cid:16) t ∞ τ i (cid:17) < R e . in this case they are no longer cofluent, the nullification of the biological rate wouldappear during development, and, as a result, the slowest biological rhythm would appear duringsleep of young individuals (figure 3c and 3f).This case analysis has an actual correspondence with empirical data for the first two cases (see forexample [HLRS64, Cra83]). We believe that theoretically biological time should be always cofluentso that the third case should never be realized. Indeed, the existing data, which are mostly givenfor humans, confirm that case does not hold (young individuals have slow rhythms, during sleeptypically, which are faster than adults slow rhythms).It would be nice that our theoretical deduction, excluding, like in physical reasoning, the thirdmathematical possibility as meaningless, were empirically confirmed in large phyla. Conversely itwould be also interesting if this theoretical derivation leads to the discovery of species where also thethird case is realized. When we consider organisms with different adult masses ( W f ), we obtain a variation of τ i according tothe scaling relationships ( τ i ∝ W / f ), whereas ω e does not change. As a result, this change correspondsto a dilatation of the ( t ) axis (as far as f is concerned) whereas the physical rhythm modifies thegeometry of biological time because the variations it triggers are anchored to the physical value ω e (seefigures 2a, without physical rhythm, and 2b, with physical rhythms.).Then, it is the interplay between physical rhythms and biological ones that breaks the symmetry(by dilatation) between organisms of different (adult) masses that have the same temporal invariants(most mammals for example). As a result, in this situation, the physical conditions can be seen asconstraints or frictions on biological temporal organization. Our point of view can be compared to thedimensionless time in [WB05], but they only consider the autonomous aspect of biological time, thusnot considering this important interplay.An other way to illustrate these aspects is to count the lifelong number of iterations of cycles: forbiological cycles, this number does not vary much when considering different species, whereas it isstrictly proportional to life span for physical ones. Let us first introduce informally the applications we will hint to in this section, where the data areobtained from the medical references in place. Our approach to biological time leads naturally, aswe will further specify, to a representation by a cylinder whose radius is proportional to the cardiac rate . If we assume that n heartbeats yield a complete rotation around the cylinder, then a faster heartrate would appear as a circular outgrowth (a sudden increase in the radius). In this representation, ahealthy individual has a complex cardiac dynamics during the day, with frequent rhythms’ accelerationsof varying length (from seconds to many hours). This shows up in the figures by the many circularoutgrowths of different radii. On the contrary, an individual with an artificially regulated pace (with11 pacemaker, say) gives a relatively smooth cylinder. The last figure below corresponds to a suddencardiac death, without particular symptoms.Of course we do not provide a theoretical determination of spontaneous biological rates variability,but just a geometrical representation . As a matter of fact in our framework, it is quite straightforwardto explore the structure of biological rhythms and of their variations. More precisely, we can easilyand effectively represent raw datas (for example the series of “beat to beat” interval over time). Asa result, we obtain more than a qualitative schema: it is a theoretical grounded representation of the“anatomy” (and pathological anatomy) of biological time. First we need to see how we can use scalesin our framework.
If we want to consider n iterations of the compacified time θ as an iteration of an other compacifiedtime ˜ θ we obtain ˜ θ = θn and ˜ s τ i = s τi n , then by some sort of renormalization using the principle ofconstant speed for the compacified time, one has: (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ −→ T τ i ( t, θ ) ∂θ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = ˜ R i n = Cst (9)So ˜ R i = R i n . This result is exactly (modulo a global dilatation of the ( t (cid:48) ) and ( z ) axis by a factor n )what we obtain if we construct directly our system at the level of n iterations.Figure 4: Renormalization and principles of variability representation.
Here, we consider ∂ −→ T τi ( t,θ ) ∂t andwe renormalize the compacified time by n = 10 . A change of speed for the iteration m of the originalcompacified time appears as a sharp contrast between this iteration and its neighbors: iteration m − , m + 1 , m − , m + 10 . As a result, if there is a coherence for 10 successive iterations, we obtain afully circular outgrowth or contraction (for an acceleration or a slowdown respectively).12 a) A global view (2 days).(b) Night, groups of 200 beats (c) Day, groups of 200 beats(d) Night, groups of 600 beats (e) Day, groups of 600 beats Figure 5:
Comparison of the situations during sleep and wake.
The point to notice here, is that thestructure tends to become a regular cylinder during night at high scales, whereas the wake is alwayscomplex. (Sample s20011 from The Long-Term ST Database, [GAG + .4.2 Rate Variability If we look at the function obtained by taking the derivative of −→ T τ i ( t, θ ) with respect to t , we obtain: ∂ −→ T τ i ( t, θ ) ∂t = t τ i τ (cid:48) (cid:16) t − t b τ i (cid:17) − R e τ i sin ( ω e t ) − πR i s (cid:48) τ i ( t ) sin (2 πs τ i ( t ) + θ ) R e τ i cos ( ω e t ) + 2 πR i s (cid:48) τ i ( t ) cos (2 πs τ i ( t ) + θ ) (10)Here, instantaneous heart rate, πs (cid:48) τ i ( t ) , appears directly as the radius of compacified time, (which hasthe physical dimension of a frequency now).If the experimental time of each heartbeat is given in a list ( t ( m )) ≤ m ≤ M , we obtain a discreteempirical version of ∂ −→ T τi ( t,θ ) ∂t , renormalized by n : ∂ −→ ˆ T τ i ( m ) ∂t = t ( m )ˆ A − ˆ R sin ( ω e t ( m )) − π nt ( m +1) − t ( m ) sin (cid:0) πmn (cid:1) ˆ R cos ( ω e t ( m )) + 2 π nt ( m +1) − t ( m ) cos (cid:0) πmn (cid:1) (11)where ˆ A is an estimation of nτ i τ (cid:48) (cid:16) t − t b τ i (cid:17) which may be soundly considered constant during the few daysof the measure. ˆ R is an estimation of n R e τ i . Both of these values are estimated by using equation 5 and ( t ( m )) ≤ m ≤ M . We obtain a 2-dimensional structure by using triangles between adjacent points, thatis to say for m ≤ M − n − , the triangles ( m, m + 1 , m + n ) and ( m, m + n, m + n + 1) . It is worthmentioning that this approach allows to obtain an empirical version of −→ T τ i ( t, θ ) too.The renormalization by n allows to observe directly the correlations between n consecutive heart-beats (a full circle) and the contrasts between a group and its neighbors (see figure 4), thus discrimi-nating easily between the sleep situation (no correlations wider than (cid:39) heart beat) and the healthywake state (correlations at each scale). The latter is indeed characterized by a succession of randomlyspaced outer circle (see figure 5).Moreover this representation may be useful to study cases of heart diseases and even aging, sincethis situations are characterized by an alteration of heart rate variability. We illustrate this alterationin cases of sudden cardiac death in figure 6 computed with datas from the The Sudden Cardiac DeathHolter Database, see [GAG + • Figure 6a is an example of a healthy case, which is characterized by a complex temporality duringwake. • In figure 6b, (intermittent) pacing leads to an excessively regular cylinder, with few heart ratevariability. • Atrial fibrillation in the figure 6c (a kind of arrhythmia, see comments in figure 6) leads to an“hairy” structure, which represents a strong short term randomness (few correlations betweensuccessive heartbeats). • Last but not least the figure 6d is not associated with a specific diagnosis (put aside suddencardiac death at time ) but it clearly shows a very simpler structure than the healthy case.Our approach allows to discriminate all these various cases by rather striking geometrical differ-ences. Wavelet analysis is often used for the same purpose, but this approach is based on a massivereorganization of datas, through a decomposition in various components, whereas we only perform ageometrical and synthetic composition of them. 14igure 6:
Comparison between a healthy situation and cases of sudden cardiac arrest. (a) Healthy case,cf figure 5. (b) Female aged 67 with sinus rhythm and intermittent pacing. (c) Female, 72, with atrialfibrillation. (d) Male, 43, with sinus rhythm. (The data are from samples 51, 35 and 30 from TheSudden Cardiac Death Holter Database, see [GAG + ( τ ) , its angles with the horizontal ϕ ( t ) and its gradients tan( ϕ ( t )) The central line ( τ ) , see figure 1, is the “result” of the various components (physical time, external andinternal rhythms) and it is supposed to refer to a “physiological” time associated to the evolution ofthe organism over the course of its life. In order to better understand the different chronological partsof life, this “axis” may be decomposed in distinct segments, each being characterized by their angle, ϕ , with regard to the abscissas (the ϕ angle under consideration then becomes that of the tangent),connected by zones with a fast curvature around specific times ( t , t , t , . . . ) . We will in particulardistinguish five parts (with unequal lengths).I Around t (which would correspond to the fertilization of the egg that will form the organismor to a mutation which generates a new species), a new segment begins with a very large angle( ◦ for example) and consequently with a very high gradient. This segment will correspond to embryogenesis .II Around t , there occurs a first curvature of the axis in order to initiate a segment of which theangle (and the gradient) still remains high (at ◦ , for example). Time t would correspond tobirth and the following segment to growth (development).III Around t , we would have a new curvature generating a medium sized angle ( ◦ for example)with a gradient approaching ; t would correspond to the apparition of the reproductive faculty(age of puberty ) and to the entering into the phase of adult maturity .IV Around t , we would have another curvature generating a small angle segment ( ◦ for example)with a weak gradient; t would correspond to the period of loss of fecundity (menopause, eventual At germination, for plants. At the moment of flowering or of fruit-bearing, for plants. and to the beginning of aging as such.V Around t the axis becomes horizontal ( ϕ = 0 , tan( ϕ ) = 0 ) and is definitely broken; t representsthe moment of death .Concerning the various durations (namely that of the life span t - t ), we know by the abovementioned laws of scaling generally encountered in biology, that these durations scale according to theorganism approximately by W / f , where W f is the mass of the adult organism.If we now consider v t = tan( ϕ ( t )) as being the “speed” of evolution of the physiological time ( τ ) with regard to the physical time t , we would make the following remarks which motivate the variousgradients of ( τ ) : • between t and t this speed is very high: initial cell divisions, morphogenesis, setting in of thefirst functionalities; • between t and t , the speed remains high; it corresponds to growth, to development, to thecompletion of the setting in of functionalities, to a high metabolism; • between t and t the speed is moderate; it corresponds to the regularity of the metabolicreactions, of cellular renewal, etc., that are characteristic of adult age; • between t and t , the speed is low: lowering of the metabolic rate, of cellular regeneration, ofactivity; this corresponds to aging; • after t the speed is null: it is the death of the organism. C e : its thread p e , its radius R i In our qualitative analysis (see 1) we have a cylinder of revolution C e , with a radius R i , which is windedas a helix having a thread of p e around the ( τ ) axis, without touching this axis but faithfully followingits changes of direction.The thread p e of this helicoidal cylinder can be assimilated to a period; it corresponds to the external cyclical rhythms imposed upon the organism by its environment (annual, lunar, circadiancycles, for instance, see §.2.2(EXT)), which are independent physico-chemical rhythms that we havetaken into account in the first paragraph; they are essentially of a physical origin and are imposedupon all organisms exposed to them. The R i = 0 case will be evoked below. C i on the cylinder and its thread p i This circular helix C i , with a thread p i , is winded around the surface of the cylinder C e (it is a “secondorder” helix because the winding cylinder is also helicoidal). We consider the thread of this helix (whichis also a period) to refer to the compacified time θ (the circle which generates this cylinder) introducedhere and associated to the internal biological cycles of the organism which are also independent (oralmost) from the environment; this is the case, let’s recall, for example, of cardiac and respiratoryrhythms, of the rhythms of biochemical cascades, etc. (see §.2.2(Int)). Let’s also recall that the periodassociated to these cycles also scale by W / f , at least from t (and also practically from t ).To summarize, we thus have, from a biological standpoint, in addition to the objective physicaltime t (evidently still present and relevant): • a general temporality of biological evolution ( τ ) (the axis); At the end of production, for plants. a temporality associated to the external rhythms (the helicoidal cylinder winded around this axisfrom a distance) that are characterized by the thread p e ; • a temporality associated to the internal rhythms involving a compactification of time: the helixwith a p i thread at the surface of the cylinder.We should also note that if the radius R i of the helicoidal cylinder becomes null, it will be reducedto a helix winded around ( τ ) and the internal cyclicity will tend to disappear as such (there remainsonly the external rhythms that are physical). If we may consider the general schema we have presentedto concern mainly the properties of the animal world, this last case, where R i = 0 , mainly concernsplant. In this sense, the non nullity of R i , that is, the two-dimensionality of the cylindrical surface,should be associated to the greater autonomy — the rhythms of the central systems, typically — and tothe autonomous motor capacity which the animal enjoys comparatively to vegetal organisms, the twobeing obviously correlated. Actually, the rhythms (metabolic, chlorophyllian, of action — activationof organs. . . ) of plants are often completely subordinated to the physical external rhythms.Of course, there is no clear-cut transition, no well-defined boundary between animal and plant lifeforms in particular in the marine flora/fauna. For this reason, we find the representation of the passingfrom the one to another in the form of a continuum to be adequate: the continuous contraction of thehelicoidal cylinder which tends towards being a helix, which is a line (the time of plants). The nonobservability of the difference between animal and plant, in some “transitional” cases, would correspondto an interval of biologically possible measurement, with no phase transition (of the type of life form)that is clear or discontinuous. Once the limit, the helicoidal line, is reached, even the three-dimensionalembedding space can be collapsed onto the two dimensions: the rhythm becomes the oscillation of onemeasurement (of chlorophyllian activity, for example) with regard to the axis of oriented physical time(the spiral is flattened into a sine, for example) as is the case in many periodic physical processes. t (cid:48) Let’s return now to our questioning regarding the interpretation we can give to the ordinate t (cid:48) . Ina certain sense, it is generated by the compacified fiber of the temporal rhythms specific to livingphenomena. More specifically, it is mathematically necessary as a component of the three-dimensionalembedding space of helixes produced by the direct production of the physical time t and of the compaci-fied time θ , which are, according to our hypothesis, two independent dimensions. We already hintedto a possible biological meaning of the ( z ) coordinate. Then, what could the ordinate t (cid:48) correspondto, from a biological standpoint?If we define a speed for the passing of time τ comparatively to t (cid:48) in a way that is similar to thedefinition of v t = tan( ϕ ( t )) , we will have v t (cid:48) = cotan( ϕ ( t )) ; at the inverse of v t (we have v t v (cid:48) t = 1 ),this velocity is small at first but continues to grow when t (or τ ) grows.In the case where the organism under consideration is the human being, an interpretation promptlycomes to mind. The velocity v t ’ would correspond to the subjective perception of the speed of thepassing of the “specific” or physiological time τ : at first very slow, and then increasingly rapid withaging. In such case, t (cid:48) would be the equivalent of a subjective time . One will notice that, from thequantitative standpoint, if between t and t (the area of the adult phase) we confer ϕ with the value of ◦ approximately, as we have already indicated above, the speed of the passing of time τ with regardto objective physical time ( v t ) coincides more or less with the subjective perception of the passing ofthis time ( v t (cid:48) ) (in fact, tan( ϕ ) (cid:39) tan ϕ (cid:39) ).As it is matter, here, of human cognitive judgment of the time flow, we are aware of its historicalcontingency. The remarks below, thus, are just informal preliminaries to forthcoming reflections, wherethe historicity of young vs. old age perception of time, for example, should be relativized to specifichistorical cultures and social frames. We then leave the reader to have any reflection regarding thesubjective perception of time during youth and old age. We can imagine that such thoughts will17oincide with ours, if we belong to the same “culture” (time which passes slowly while young and, later,very quickly. . . ).In what concerns organisms other than human beings, of which we do not know if they have asubjective perception of the speed of the passing of physiological time τ , it is more difficult to assigna clear status to this dimension of t (cid:48) (although certain relatively evolved species seem likely to expressimpatience, for example, or to construct an abstract temporal representation by exerting facultiesof retention and especially of protention). So would this dimension not begin to acquire a concretereality only with the apparition and development of an evolved nervous system (central nervous system,brain)? But then what of the bacterium, the amoeba, the paramecium. . . ?Actually, it may be possible to somewhat objectivize the approach by advancing a plausible hy-pothesis regarding the general character of t (cid:48) : we could consider that it is a question of a “temporality”that is associated to the “representational” dimension. Let’s explain.Since living organisms are endowed with more or less capacity for retention and protention (possiblypre-conscious “expectation”), we propose (temporarily, this is ongoing work) to base ourselves on thefollowing qualitative argument: the element of physiological time dτ is associated to the element ofphysical time dt and to dt (cid:48) by the evident relation dτ = dt + dt (cid:48) ; it stems from this that dt (cid:48) canbe written as dt (cid:48) = dτ − dt or as dt (cid:48) = ( dτ − dt )( dτ + dt ) It is then tempting to see in the first factor the minimal expression of an element of “retention”(for physiological time, relatively to physical time) and in the second the corresponding expressionof an element of “protention”. The product of the two would generate the temporality component ofa “representation” which borrows from the “past” and from the “future”, as constitutive of the flowof biological time. As all living organisms appear to be endowed with both a capacity for retention— as rudimentary as it may be — and with a protentional faculty (even more rudimentary maybe),the generality of the dimension t (cid:48) would be preserved and the “representational” capacity (at least inthis elementary sense) appears as being a property of living phenomena, see [LM11]. This property,for conscious thought, could even be extended to subjectivity in accordance, in the specific case ofthe human being, with the phenomenological analysis with which we began: dt (cid:48) would be a form, aselementary as infinitesimal, of the “extended present”, in the husserlian tradition, described by otheranalyzes, such as the coupling of oscillators in [Var99].Finally, it would be the two-dimensionality t × t (cid:48) — (physical time) × (representation time) —which would enable to mark out the temporality of living phenomena, which may be represented inthe geometrical way as we have described in this paper. 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