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, or through time averages, if we know the occurrence time of crucial events . But, the complexity matching theory of this paper allows us to establish the transfer of information from one complex network to another at the level of single realizations, for example matching between the movements of two Tango dancers. |
Note that the 1/f-variability of the spectrum is a necessary, but not a sufficient, condition to have maximum information exchange between two complex networks. This is where the present theory deviates from the early form of cybernetics. The present theory requires the existence of crucial events. |
Homeodynamics: Another important property of biological processes is homeodynamics , which seems to be in conflict with homeostasis as understood and advocated by Ashby. Lloyd et al. |
invoke the existence of bifurcation points to explain the transition from homeostasis to homeodynamics. This transition, moving away from Ashby’s emphasis on the fundamental role of homeostasis, has been studied by Ikegami and Suzuki |
and by Oka et al. , who coined the term dynamic homeostasis . They used Ashby’s cybernetics to deepen the concept of self and to establish if the behavior of the Internet is similar to that of the human brain. |
Turalska et al. , based on the direct use of the dynamics of two complex networks, studied the case when a small fraction of the units of the regulated system perceive the mean field of the regulating system. At criticality the choice made by these units is interpreted as swarm intelligence |
, and, in the case of the Decision Making Model (DMM) adopted in is associated with the index [MATH] Synchronization is observed in |
when both systems are in the critical condition [MATH] and it is destroyed if one system is critical and the other is sub-critical, or viceversa. This suggests that maximal synchronization is realized when both systems are at criticality, namely, they share the same IPL index [MATH] |
The present theory covers the complexity matching between networks with different complexity indices [MATH] . Also, the present theory is supplemented by homeodynamics, which had not been considered before. This theory should not be confused with the unrelated phenomenon of chaos synchronization. In fact, the intent of... |
METHOD To address Ashby’s challenge we adopt the perspective of subordination theory . This theoretical perspective is closely connected to the Continuous Time Random Walk (CTRW) |
, which is known to generate anomalous diffusion. We use this viewpoint to establish an approach to explaining the experimental results showing the remarkable oscillatory synchronization between different areas of the brain |
It has to be stressed that a natural choice may rest on the use of Kuramoto’s model . In fact, the model of Kuramoto affords a simple paradigm to explain synchronization of rotators, as explained in the excellent review paper of Ref. |
. We think that this popular model can be properly generalized to replace the adoption of its control parameter with the same self-control of Refs. |
that is shown to generate criticality. The theory of these papers, called Self-Organized-Temporal Crtiticality (SOTC), makes the control parameter spontaneously evolve towards a condition of fluctuation around a critical value that, interpreted as a fluctuating temperature may lead to physical effects similar to those ... |
, which affords again a natural way to explain the synchronization of rotators. However, this direction, which is left as a subject of future investigation, would make more difficult for us to explain the important role of crucial events for the realization of synchronization. For this reason in this paper we adopt the... |
, which is a generalization of CTRW , based, indeed, on activating the action of crucial events. Hereby we describe subordination to periodic and regular rotation. This theory, supplemented by the intelligence necessary to realize CTRW leads us to the central algorithmic prescription of Eq. ( ) ) and Eq. ( ): there is ... |
Consider a clock, whose discrete hand motion is punctuated by ticks and the time interval between consecutive ticks is, [MATH] , by assumption. At any tick the angle [MATH] of the clock hand increases by [MATH] , where [MATH] is the number of ticks necessary to make a complete rotation of [MATH] . We implement subordin... |
During the dynamical process the signal frequency fluctuates around [MATH] and the average frequency is changed into an effective value |
[EQUATION] This formula can be easily explained. In fact, [MATH] is the border between two distinct statistical regions, the Levy and the waiting-time PDF of the Gaussian region [MATH] where both the first and second moment of [MATH] are finite, and the average of the fluctuating frequencies is identical to [MATH] . In... |
[MATH] replaced by [MATH] . Thus, using the result of earlier work we obtain for the equilibrium correlation function exponentially damped regular oscillations. At the end of this oscillatory process, an IPL tail proportional to [MATH] is obtained. Using a Tauberian theorem explains why the power spectrum [MATH] become... |
for [MATH] . In summary, in a log-log representation, we obtain a curve with different slopes: [MATH] , to the left of the frequency-generated bump, and [MATH] , to its right. The slope [MATH] |
is a consequence of the exponentially damped oscillations. These predictions are confirmed in The spectrum [MATH] of subordinations to the regular clock motion. (a) [MATH] [MATH] (black curve), [MATH] (red curve). (b) [MATH] [MATH] (black curve), [MATH] (red curve). , which illustrates the result of a numerical approac... |
To make network-1 (S [MATH] drive network-2 (S ) we have to generalize the swarm intelligence prescription adopted in earlier work |
. This generalization is necessary because the earlier work was limited to matching of two identical networks at their criticality, and also was based on the assumption that the single units of the complex networks, in the absence of interaction, undergo dichotomous fluctuations without the periodicity imposed here. In... |
[MATH] [MATH] . In the case considered herein the number of units in a network [MATH] is constant. Using this notation (see Section we show that network-2 under influence of network-1 changes as: |
[EQUATION] where [EQUATION] To properly take periodicity into account note that the mean field in S given by [MATH] has the functional form |
[EQUATION] The mean field in S has the same periodic functional form, up to a time-dependent phase, [EQUATION] The phase [MATH] is a consequence of the fact that the units of S |
try to compensate for the effects produced by the two independent self-organization processes. The number of ticks of the S clock, [MATH] , due to the occurrence of crucial events, becomes increasingly different from the number of ticks of the S clock, [MATH] . The units of S try to imitate the choices made by the unit... |
[EQUATION] if at [MATH] no crucial event occurs, and [EQUATION] if at [MATH] a crucial event occurs. Note that the real positive number [MATH] , defines the proportionality factor left open by Eq. ( ), or, equivalently, defines the strength of the perturbation that S exerts on S |
(blue curve) drives S (red curve). Two systems are identical: [MATH] [MATH] [MATH] The connection (one directional) is realized using Eq. ( ) and Eq. ( 14 ). illustrates the significant synchronization between the driven and the driving system obtained for [MATH] , close to the values of the crucial events of the brain... |
. This result also can be used to explain the experimental observation of the synchronization of two people walking together (see Section ). |
The top panel of The spectrums of subordinations. (a) Black curve ( [MATH] ): [MATH] ; red curve ( [MATH] ): [MATH] ; blue curve: [MATH] after being connected (one directional) to [MATH] with [MATH] . (b) Black curve ( [MATH] ): [MATH] |
[MATH] ; red curve ( [MATH] ): [MATH] [MATH] ; blue curve: [MATH] after being connected (one directional) to [MATH] with [MATH] shows that S with [MATH] , is very close to the Gaussian border and adopts the higher complexity of S with [MATH] , namely the complexity of a network very close to the ideal condition, [MATH]... |
In the bottom panel of The spectrums of subordinations. (a) Black curve ( [MATH] ): [MATH] ; red curve ( [MATH] ): [MATH] ; blue curve: [MATH] after being connected (one directional) to [MATH] with [MATH] . (b) Black curve ( [MATH] ): [MATH] |
[MATH] ; red curve ( [MATH] ): [MATH] [MATH] ; blue curve: [MATH] after being connected (one directional) to [MATH] with [MATH] we see that a driving network very close to the Gaussian border does not make the driven network less complex, but it does succeed in forcing it to adopt the regulator’s periodicity. Here we h... |
. In this latter case, according to Heidegger’s phenomenology the transition from ready-to-hand to unready-to-hand makes the IPL index [MATH] depart from the [MATH] -noise condition [MATH] |
so as to reach the Gaussian border [MATH] and to go beyond it. Here the perturbation is characterized by an intense periodicity and while it does not change the complexity of the perturbed network very much, it does transfer its own periodicity. |
The theory developed herein may shed light on the crucial role of cooperation. Recent psychological research on collective intelligence |
shows that a cooperative interaction between the members of a group may improve the global intelligence of a group. To realize a condition that is close to that of Ref. |
we study the case where S is influenced by S in the same way S is influenced by S . To make this extension we have to introduce the new parameter [MATH] , which defines the intensity of the influence of S on S [MATH] |
As a result of this mutual interaction, we have [MATH] and [MATH] . When [MATH] we expect [EQUATION] The spectrums of subordinations. Black curve ( [MATH] ): [MATH] [MATH] ; red curve ( [MATH] ): [MATH] [MATH] . Blue and pink curves are the spectrums of [MATH] and [MATH] after being connected (bidirectional) with [MATH... |
, while the less complex system has a sense of relief. We interpret this result as an important property that should be the subject of psychological experiments similar to that of Ref. |
to shed light on the mechanisms facilitating the controlled exchange of information in the teaching and learning processes. The theory underlying complexity matching and therefore requisite variety, makes it possible to go beyond the limitation of the earlier work on complexity management, as illustrated in Section |
The term “intelligent” that we are using herein is equivalent to assessing a network to be as close as possible to the ideal condition [MATH] , corresponding to the ideal [MATH] noise. The spectrums of subordinations. Black curve ( [MATH] ): [MATH] [MATH] ; red curve ( [MATH] ): [MATH] [MATH] . Blue and pink curves are... |
Note that in the same sense two very intelligent networks are the brain and heart that when healthy share the property of a [MATH] being close to [MATH] . The argument presented herein therefore provides a rationale for (an explanation of) the synchronization between the heart and brain time series |
showing that the concept of resonance, based on tuning the frequency of the stimulus to that of the network being perturbed, may not be appropriate for complex biological networks. Resonance is more appropriate for a physical network, where the tuning has been adopted over the years for the transport of energy not info... |
, are the subject of appraisal and the present results may contribute to making therapeutic progress by establishing their proper use. |
Supporting Information This section affords an example of the application of the theory developed herein to the analysis of experimental data. We focus on the close connection between Fig. 2 of this paper and Fig. 3 of Ref. |
. For reader’s convenience we illustrate this connection with the help of Experimental walking synchronization. These results have been derived with permission from Ref. |
. The top panel shows two distinct walking trajectories. These are two human subjects trying to walk together. The bottom panel shows the same trajectories so as to emphasize their synchronization. |
This figure is the result of the real experiment of Ref. and it should be compared to the qualitatively similar Time difference between the events of two identical systems connected back to back, [MATH] [MATH] [MATH] obtained with the theory of this paper. |
We obtain Time difference between the events of two identical systems connected back to back, [MATH] [MATH] [MATH] using Eqs. (2-5) of the text and hereby we afford details on how to derive these important equations. We use numerical results of the same kind as those illustrated in Fig. 2 properly modified to connect t... |
more evident we adopt the same prescription as that used by Deligniéres and his co-worker and interpret the time interval between consecutive crossings of the origin, [MATH] , of Fig. 2 as the time duration of a stride. We evaluate the mean stride duration and for both the driven and the driving, for any stride we plot... |
3.1 Group intelligence Although subordination theory does not explicitly depend on the interaction between different units with their own periodicity, S is driven by S with a prescription inspired to create a swarm intelligence |
. At a given time the units of the driven systems look at the driving system and according to its state increase or decrease the phase of the driven system as described in the Methods section. |
The single individuals of the complex network may have only the value [MATH] , cooperation, or [MATH] , defection. We introduce the angle [MATH] to take periodicity into account and interpret [MATH] as the ratio of the difference between the number of cooperators and the number of defectors to the total number of units... |
[EQUATION] where the probability of making a transition from the state down to the state up in S is given by [EQUATION] The form of Eq. ( 12 ) is due to the fact that this probability is the product of the probability of finding a unit in the driven system in the down state by the probability of finding a unit in the d... |
[EQUATION] Let us plug Eq. ( 12 ) and Eq. ( 13 ) into Eq.( 11 ). We obtain [EQUATION] which is the important prescription of Eq. ( ). |
3.2 Walking together To facilitate appreciation of the similarity between the complexity matching prescription of this paper and the walking synchronization of Ref. |
, we invite the readers to look at the experimental results of Experimental walking synchronization. These results have been derived with permission from Ref. |
. The top panel shows two distinct walking trajectories. These are two human subjects trying to walk together. The bottom panel shows the same trajectories so as to emphasize their synchronization. . The real data are not available to us, and we use surrogate data instead. These surrogate data are derived from the nume... |
. The top panel shows two distinct walking trajectories. These are two human subjects trying to walk together. The bottom panel shows the same trajectories so as to emphasize their synchronization. proves the efficiency of the complexity matching approach of this paper. |
3.3 Beyond Complexity Management Complexity management is difficult to observe, since it is based on ensemble averages, thereby requiring the average over many identical realizations |
. In the case of experimental signals of physiological interest, for instance time series relating to brain dynamics, taking the ensemble average is not possible. Recently a procedure was proposed |
to convert an individual time series into many independent sequences, so as to again have recourse to an average over many realizations. This procedure, however, requires knowledge of time occurrence of crucial events. The theory developed herein makes it possible to evaluate the correlation between the driving and the... |
does not affect the power index [MATH] of the interacting complex networks, the present theory, as shown by The spectrums of subordinations. (a) Black curve ( [MATH] ): [MATH] ; red curve ( [MATH] ): [MATH] ; blue curve: [MATH] after being connected (one directional) to [MATH] with [MATH] . (b) Black curve ( [MATH] ): ... |
[MATH] ; red curve ( [MATH] ): [MATH] [MATH] ; blue curve: [MATH] after being connected (one directional) to [MATH] with [MATH] , affords important information on how the cooperative interaction makes the unperturbed values of [MATH] change. |
In Dependence of [MATH] (as a measure for complexity matching) on the periodicity of the drive and driven systems. [MATH] [MATH] the maximum value of the cross correlation function, [MATH] , between the driving and the driven. This figure shows that a significantly large frequency mismatch strongly reduces the intensit... |
Dependence of [MATH] on the complexity index of the drive and driven networks. [MATH] [MATH] shows the effect of changing [MATH] and [MATH] on [MATH] , while keeping the frequencies [MATH] identical. |
Complexity, Information and Conclusions In the recent literature on self-organization, see, for example, Gershenson and Fernández |
, emergence of complexity is interpreted as corresponding to information reduction, while emphasizing Ashby’s concept of homeostasis. Variety increases with a complex system performing multitask actions and decreases with a complex system focusing on a single task |
. More recent work confirms this property in sociological systems while it is well known that it holds true for physiological processes |
. The hypothesis of self-organization has been known and used in biology for nearly half a century (see also Chapter 5 of Eigen’s important book |
). The theoretical approach to the complexity matching presented herein may afford a unifying view accounting for the main properties emphasized by this literature while adapting the homeostasis perspective of Ashby |
to the concept of homeodynamics. Consider the four distinct properties stressed in the current literature: Information reduction, requisite variety, multitasks, homeostasis. |
Information reduction The entropic approach used to deal with crucial events is the Kolmogorov-Sinai (KS) entropy [MATH] , which is well described by the formula |
[EQUATION] where [MATH] . Eq.( 15 ) indicates that the KS entropy vanishes at [MATH] and it remains equal to [MATH] in the whole infinite interval [MATH] [MATH] ). Allegrini et al. |
noticed that [MATH] , corresponding to [MATH] , is the condition of total randomness, namely, the case where an infinitely large amount of information is necessary to control the system. The condition [MATH] , corresponding to [MATH] , makes the sequence of crucial events compressibile, namely, it reduces the amount of... |
The recent generalization to the mechanism of self-organized criticality given by SOTC generates crucial events with [MATH] , and, albeit a form of self-organization yielding values of [MATH] in the interval [MATH] is not yet known, we make the plausible conjecture that complex processes that are experimentally proven ... |
had to modify Eq. ( 15 ) leaving this expression unchanged for [MATH] and making it increase from the vanishing value with [MATH] . Actually [MATH] entropy is a Lyapunov coefficient and Korabel and Barkai defined the Lyapunov coefficient for [MATH] , by comparing the departure between two trajectories moving from very ... |
Requisite Variety Ivanov et al. noticed that the healthy heartbeats have a variability that makes it impossible to adopt the conventional method of analysis of anomalous scaling based on the stationary assumption. Consequently they made the assumption of a scaling fluctuation that led them to adopt a multi-fractal appr... |
. Allegrini et al. examined the same patients studied in using the crucial events defined herein and found that healthy patients have a [MATH] very close to the value [MATH] , which makes the [MATH] entropy vanish. They also conjecture that a self-organization generating crucial events may also be the generator of mult... |
Of remarkable importance for the requisite variety issue is the work by Struzik et al , emphasizing the transition from [MATH] noise to [MATH] noise as a manifestation of variability suppression. Healthy heart physiology is based on the balance between the conflicting action of the sympathetic and parasympathetic nervo... |
yields [MATH] . We examined the case of [MATH] moving in the interval [MATH] using subordination to regular oscillatory motion, a phenomenological way of combining crucial events and periodicity. We believe that SOTC can be extended to this condition, and hope that future work may realize this important goal. We see th... |
Monotasking versus Multitasking : The theoretical perspective adopted herein affords an efficient way to approach this problem, while suggesting an interesting approach to cognition. In a recent paper Gershenson |
addresses the important issue of the connection between cognition and information and writes: “Just like Buddhist philosophy, information theory and current cognitive science are pointing towards a worldview not centered on objective phenomena (studied traditionally by physics), but centered on information, which can r... |
. These authors rest on Buddhism to create a bridge between consciousness as a phenomenon in the operational architectonics of brain organization and quantum mechanics. The earlier work of these authors led to the discovery of rapid transition processes (RTP) that have been studied in |
, and found to be crucial events with [MATH] , but very close to [MATH] . For this reason, we are convinced that the theoretical approach adopted herein may help to build the bridge between West and East that the authors of |
and are trying to establish. It is very encouraging to notice that Tuladhar et al have recently found that meditation has the surprising effect of enhancing heartbeat coherence generating effects qualitatively similar to those illustrated in Fig. 1 of this paper, thereby leading us to interpret meditation as a mental p... |
Homeodynamics : The important results established herein are based on the action of crucial events, which are a manifestation of temporal complexity, thereby explaining why we replace Ashby’s homeostasis with homeo-dynamics. |
Finally, we conclude this paper stressing that the surprisingly accurate synchronization of the walking together process ought not to be confused with either chaos synchronization or resonance. In fact, chaos synchronization requires finite Lyapunov coefficients and resonance requires frequency tuning. Complex systems ... |
and . Thus, our prediction that the walker with [MATH] close to [MATH] attracts the [MATH] of the walker close to the Gaussian region [MATH] can be interpreted as a transmission of multifractality from the healthy to the sick walker in a surprising agreement with the recent experimental result of |
Acknowledgments The authors thanks Dr. Herbert Jelineck drawing our attention to Ref. , PG thanks ARO for financial support of this work through grant W911NF1901. |
# Source: arxiv 1806.09057 # Title: In-situ Stochastic Training of MTJ Crossbar based Neural Networks # Sections: all # Downloaded: 2026-03-03T02:18:06.859092+00:00 |
In-situ Stochastic Training of MTJ Crossbar based Neural Networks Abstract. Owing to high device density, scalability and non-volatility, Magnetic Tunnel Junction-based crossbars have garnered significant interest for implementing the weights of an artificial neural network. The existence of only two stable states in M... |
1. Introduction Deep Neural Networks (DNNs) have become a popular choice for tasks such as image classification, face recognition, and Natural Language Processing. This has however been at the cost of massive computations on von Neumann architectures exhibiting high energy and area requirements (et al., 2012a . The eme... |
Attempts have been made towards dedicated hardware designs and realization of the synaptic weights (and neurons) of a Neural Network (NN) by using CMOS transistors in an analog fashion (et al., 2010b ; but these have met with challenges of scalability and volatility. Parallel research work has focused on using post-CMO... |
Another choice is the Magnetic Tunnel Junction (MTJ), an emerging binary device (since it has 2 stable states) which has shown its potential as storage elements and is a promising candidate for replacing CMOS in memory chips (et al., 2013b . Its non-volatility and scalability makes it a particularly lucrative choice fo... |
Continuous weight networks can be simplified into discrete weight networks without significant degradation in classification accuracy while achieving substantial power benefits (et al., 2016f . The use of discrete weight networks, such as BinaryConnect (et al., 2015d and in (et al., 2016e , also stems from the challeng... |
Obtaining optimal weights for a binary network in software can be impractical because its discrete nature requires integer programming. Also, when physically realizing an NN on hardware, the underlying device variations can have a substantial impact on the model accuracy, and need to be accounted for in the training pr... |
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