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Two Agent Coherent-Stressful Interaction HRV based Simulation Model

Jeffery Jonathan Joshua (ישוע) Davis, Florian Schübeler

The Embassy of Peace, Whitianga, New Zealand

joshua_888@yahoo.com, florian@theembassyofpeace.com

In the last decade a body of research has been published on psychophysiological coherence measured via Heart Rate Variability (HRV) and its associated benefits for general wellbeing. However, there is significantly less literature describing the dynamics between two (2) or more agents (people) interacting and affecting each other's inner states, which would also be reflected in HRV measures. This paper addresses this challenge by the use of stochastic simulation based on research data in order to describe the dynamics between two (2) agents during a horizon of several months in order to estimate the psychophysiological transformations of each agent after several interactions. This we conjecture will provide a better understanding about how to manage relationships when the parties involved are equipped internally with different levels of personal mastery in upholding coherent states for long periods of time. Ideally, after understanding the scenarios analysed the reader should have an insight into how to better support individuals in the mastery of psychophysiological coherence.

joms | 2019 | 1(32)
Reviewed By: Shruti Jain | Communication Engineering Department, Jaypee University of Information Technology, Solan, Himachal Pradesh, India.
Mr. Shamneesh Sharma | Associate Professor, Department of Computer Science & Engineering, Alakh Prakash Goyal Shimla University, Shimla (H.P.) INDIA
Dates: Submission: 08-Jan-2019 | Acceptance: 06-Jun-2019 | Publication: 07-Jun-2019

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INTRODUCTION

Recent studies have shown the benefits of mastering HRV associated coherent states, something termed psychophysiological coherence (McCraty, 2002), (McCraty & Tomasino, 2004), (McCraty & Childre, 2010), (McCraty & Shaffer, 2015). However, there is significantly less literature describing the interactions between agents (people) that affect each other's inner states and HRV (McCraty et al., 1998).


Here we explore what could happen when two (2) agents (let us say good old Bob and Alice) meet and interact for a period of time where the state of one (1) person influences the state of the other. We describe in a simplified manner, akin to our model, in what kind of states a person can exist (stressed, relaxed and coherent) and how an interaction with someone else can cause a transition from one (1) state to another in one (1) or both agents. We developed a stochastic simulation model (Law & Kelton, 1991) based on Markov Transition Probability Matrices (TPM) (Ross, 1985), (Ross, 1983) that describe the baseline individual dynamics for each agent and then as they interact in time we adjust their TPM according to certain rules.


The process Xt describes the evolution of the psychophysiological stressed, relaxed or coherent states (0, 1 & 2 respectively) usually associated with scores St (-1, 1 & 2 respectively) as described in (Davis & Schübeler, 2019), (Davis et al., 2019), (Davis & Schübeler, 2019a), (Davis et al., 2019a). However, in this work we will focus on Xt only. Finally, we run the model long enough to understand the dynamics and transformations of each agent based on both their initial baseline activity and the successive transformation until they reach a new tendency with its associated steady states. The reader must note that one (1) individual could have an initial coherent baseline while the other would show a stressful one (1) and depending on: (1) how resilient and set in their baseline state they are and (2) their capacity to regenerate, both may end up stressed, relaxed or coherent, better or worse than their original baseline. Finally, we present the reader with some conclusions and future perspectives concerning the potential to understand and manage psychophysiological states to the benefit of the individual, the community and large population groups.


SYSTEM ANALYSIS AND MODELS

A. Scenario Descriptions

In this section we first consider all the possible interactions between Bob and Alice where any of them could be in a coherent, relaxed or stressful tendency represented by the colours green, blue and red. Also, we introduce another dimension, which describes how strong they are at staying in their tendency, meaning how hard or easily can they be influenced to make a transition into another type of tendency. Initially it would appear that we should model a minimum of nine (9) possible types of interactions as described in Table I; that is without including the strength-dimension. However, with little analytical effort we can cut the number of scenarios to one (1) where one (1) agent is in a green baseline and the other in red.


Table I. | Possible individual states for interactions between Bob (B) and Alice (A)

This follows the rationale that modelling the two (2) agents in red, blue or green would lead to no change in either of them. Blue tendencies are rare generally and they are by nature transitional towards green or red, the coherent or stressed psychophysiological states. Therefore any interactions involving blue baseline tendencies are unnecessary to model.


Because of that we are left with the scenarios to model when Bob is in green and Alice is in red in baseline and vice versa. However, this is also redundant since we can think impersonally in terms of modelling one (1) agent in green and the other in red, which reduces the problem to one (1) type of interaction from which we can derive a set of scenarios for different values of the strength-dimension mentioned above.


Concerning the strength-dimension both Bob and Alice are assigned values that reflect their degree of strength or the degree to which they can be influenced towards a change in tendency. This is explained in detail in the following section on mathematical modelling.


B. Mathematical Modelling

In this section we introduce the reader to a set of mathematical tools and models in order to characterise the system and create the foundations for the simulation model.


Let us remember that a Markov process (Allen, 1978) can be described as a set of transitions from one (1) state to another where Xt describes the evolution of the process in discrete time t for a particular system where the TPM (P) of the system for three (3) states of X: {0, 1 & 2} is represented as:



where Pij is the probability of making a transition from state i to state j between t-1 and t. Particularly in our case, time t is a discrete variable and hence the model presented here is a discrete Markov process or more precisely, a discrete-time Markov chain on a countable state space.


First we describe the different tendencies in terms of Markov TPM, where Bob, from now on labelled as 'B', presents a green tendency described by the following baseline and initial (t=0) TPM B0:



while Alice, from now on labelled as 'A', presents a red tendency described by the following baseline and initial (t=0) TPM A0:



Following we describe the strength-dimension as weight parameters, αB and βB for agent B and αA and βA for agent A, where:


βB = 1 - αB and βA = 1 - αA           (1)

Let us imagine that αB = 0.8 and βB = 0.2 while αA = 0.3 & βA = 0.7. This would mean that B would more likely stay in his tendency while A would more likely shift to B's tendency. The rule that governs this change is prescribed as follows:


Bt = αB * Bt-1 + βB * At-1           (2)


At = αA * At-1 + βA * Bt-1          (3)


For our particular case this would look as follows:


Bt = 0.8 * Bt-1 + 0.2 * At-1       (4)


At = 0.3 * At-1 + 0.7 * Bt-1       (5)


When applying this formula iteratively from t=0 to t=1, then we would obtain:

B1 = 0.8 * B0 + 0.2 * A0       (6)

A1 = 0.3 * A0 + 0.7 * B0       (7)


and after this set of transformations we would obtain, for t=1, the following TPM B1 and A1:




If we compute the steady states for all these matrices we could easily trace the evolution of the interactions by also computing the expected values of each Markov chain for each time step. The equations for the limiting stationary probability distribution π = (π0 π1 π2), where the condition π = π*P is met and the expected value for a three (3) state Markov chain with TPM (P) are as follows (Allen, 1978):


Let πj(t) = P (Xt = j)      for j = 1, 2, and 3                                        (8)


πj = ∑i πi * Pij      for j = 1, 2, and 3                                               (9)


πj = lim t➝∞ πj(t) = lim t➝∞ (P (Xt = j))t       for j = 1, 2, and 3      (10)


The limiting probabilities πj must meet the condition that π0 + π1 + π2 = 1 and the expected value (E) of the process is computed as E(S) = 0 * π0 + 1 * π1 + 2 * π2 or just E(S) = π1 + 2 * π2. It is important to note that an estimate of π can be computed by Pt where t➝∞. Every row of this matrix is equal to every other row and comprises π0, π1, and π2. In our case, the limiting probabilities for A0, A1, B0 and B1 are displayed in Table II.


Assuming Bob and Alice meet weekly, for example, we can observe that Bob is slowly deteriorating, even though he is more likely to be found in green. Alice on the other hand has improved dramatically and shifted tendency from being more likely to be found in red, to more likely be found in green. This is because Alice can be more influenced by Bob than Bob can be by Alice. If Bob is unable to recover his baseline until the next encounter happens then these tendencies will continue, Bob deteriorating and Alice improving until some form of middle ground steady state or equilibrium is reached.


Table II. | Limiting probabilities for the TPM of A and B for t=0 and t=1

The expected values for Xt in each step t=0 and t=1 for Bob and Alice are: E (XA, t=0) = 0.1507, E (XA, t=1) = 1.3134, E (XB, t=0) = 1.7345 and E (XB, t=1) = 1.4690. Note the dramatic improvement of Alice from 0.1507 to 1.3134 between t=0 and t=1. Also note that Bob is still better off or more coherent than Alice, however with values very close to Alice in t=1.


C. Scenarios, Simulations and Results

In this section we describe the scenarios to be simulated with the aid of a stochastic simulation model developed for this purpose and we also report the simulation results. We designed four (4) types of scenarios as follow: (a) scenario without recovery, (b) scenario with full recovery after every encounter, (c) scenario with delayed recovery and (d) scenario with diminishing recovery. It is important to note that this model could apply to any horizon and time units, such as minutes, hours, days, months and years depending on the type of agent dynamics to be studied. Some examples are: (a) an interaction between two (2) agents discussing an issue where we model a horizon of one (1) hour and monitor the system every minute, (b) daily yoga sessions where we monitor the system every day for a horizon of a week and (c) a marital relationship where we monitor the system every month for a horizon of a year. For the scenarios presented here we simply use general units of time t. It is also important to note that we have applied our models for a limited amount of scenarios with a limited range of assumptions for the paper restrictions, however, we could easily adjust the assumptions and create a diversity of situations and scenarios that could be modelled.


The following list gives a description of the simulated scenarios:


Scenario a: describes a situation where Bob has a baseline with a marked green tendency while Alice has a baseline with a marked red tendency. Also, Bob and Alice have strength-dimension parameters of {αB = 0.8, βB = 0.2} and {αA = 0.3, βA = 0.7}, which implies that Bob will influence Alice strongly whereas Alice will only influence Bob weakly.
Scenario b: is similar to Scenario a, however in this scenario Bob recovers completely to his original baseline after every encounter with Alice, in other words, he never deteriorates.
Scenario c: is similar to Scenario a, however Bob recovers and deteriorates in cycles. Here we introduce four (4) variants as Scenario c1, c2, c3 and c4 for encounters with strong-strong (c1), strong-weak (c2), weak-weak (c3) and weak-strong (c4) strength-dimension parameters. All four (4) variants should be interpreted in a Bob-Alice order.
Scenario d: is similar to Scenario a, however Bob deteriorates in terms of the strength-dimension parameters with a mild exponential decay. Here we introduce two (2) variants as Scenario d1 and d2 for a slow and fast deterioration respectively.


Figure 1 shows the results of the simulation for Scenario a, where we observe how Alice improves at the expense of Bob and they stabilise to a value of 1.37 for both E (XA) and E (XB). This value is still associated to a green tendency, however smaller than the original value of 1.8 for Bob, which is very close to the maximum value of 2 (the green state). It takes around five (5) encounters (one per week, for example) in order for them to reach the steady state.


Figure 1 | displays the development of E(X) for Alice (dashed line and triangular markers) and Bob over time. While Alice improves, Bob slightly deteriorates.

Figure 2 shows the results of the simulation for Scenario b where we can clearly observe that Bob is always at his maximum expected value E(XB) of 1.735 and Alice improves without Bob deteriorating.


Figure 2 | displays the development of E(X) for Alice (dashed line and triangular markers) and Bob over time. While Alice improves, Bob remains stable.

This could be associated to, for example, the fact that Bob meditates, exercises and sleeps properly after each encounter and until the next one (1) and therefore he fully recovers to his baseline.


In Figure 3 we show a comparative graph of the simulations associated to Scenario c1, c2, c3 and c4.


Figure 3 | displays the development of E(X) for Alice (dashed lines) and Bob over time for four (4) different scenarios. While Alice always improves, Bob goes through oscillations of temporary deterioration. Scenario c2 (Figure 3b) displays the smallest oscillations and Scenario c3 (Figure 3c) the strongest oscillations.

Here we can observe for the four (4) variants that Alice will always reach the maximum expected value set by Bob, E(XB) = 1.735 regardless of the strength-dimension parameters associated to each. However, there are some important distinctions to highlight:


1. When Bob and Alice are both strong in their tendencies then Alice will improve with minimal cost to Bob where Bob will mildly oscillate. They will both reach steady states after around forty (40) encounters.
2. When Alice is weak and Bob is strong then Alice will improve very fast with very minimal cost to Bob, where Bob will barely oscillate and they will both reach steady states after around eight (8) encounters.
3. When Alice is strong and Bob is weak then Alice will improve very slowly with very great cost to Bob and Bob will oscillate dramatically until they both approach steady states after around 120 encounters.
4. When Alice and Bob are both weak then Alice will also improve slowly with a very significant cost to Bob and Bob will oscillate strongly until they approach steady states after around thirty-five (35) encounters. Note that even though Bob displays oscillations the time when they reach steady states is significantly shorter than when Alice is strong.


In Figure 4 we present a comparison of the two (2) variants: Scenario d1 and Scenario d2, where Bob loses strength-dimension as time goes by and that means that he becomes gradually more susceptible to be influenced by Alice.


Figure 4 | displays a comparison of Scenario d1 and d2. In Scenario d2 Bob and Alice stabilise in higher values of X, since Bob was able to uphold a greater strength, while supporting Alice. In Scenario d1 Bob had a lower strength level and therefore both stabilise in lower values of X. However, in both scenarios Bob deteriorates and Alice improves.

The reader must note that when Bob upholds a greater strength-dimension, he is able to uplift Alice to a steady state of higher value. The opposite is also true. In both scenarios, they reach the steady state in around four (4) encounters. (Denney, 2008) (McCraty et al., 1998).


D. Some further considerations

In this section we present the reader with two (2) simulations for Scenario a and a*, where we show the cumulative scores CSt associated to the scores St based on the stochastic evolution of the process Xt as treated in (Davis & Schübeler, 2019), (Davis et al., 2019), (Davis & Schübeler, 2019a) and the limiting probabilities π0, π1, and π2 for the initial (baseline) TPM as well as the combined effect of the evolving TPM. Scenario a* introduces a variation on Scenario a as follows:
{αB= 1, βB = 0} and {αA= 0, βA = 1}


<
Figure 5 | displays Scenario a on the left and Scenario a* (with strength tendency variations) on the right, where both Alice and Bob are very strong in their tendencies. The top graphs show the limiting probabilities for both scenarios and the bottom graphs show the CS vs. the Ideal CS, for both Alice and Bob.

In Figure 5 (bottom graphs) we observe the Cumulative Score (CS) for two (2) simulations:


=> Scenario a*: The CS for Bob and Alice vs. the Ideal CS for initial (baseline) TPM as if they never affected each other (bottom right).
=> Scenario a: The CS for Bob and Alice vs. the Ideal CS for the compound effect of all the evolving TPM as the interactions between them happen over time (bottom left).


As we can clearly observe in Figure 5 (bottom right) Bob's CS evolves very close to the Ideal CS (always in green) since he has a strong green tendency and baseline while Alice on the other hand displays CS very close to zero (0) most of the time since she has a red initial baseline. However, for the case where they do affect each other, we observe how Bob displays lower CS farther from the Ideal CS while Alice has improved showing CS closer to the Ideal CS, very similar to Bob's. These results reflect the influence of Alice and Bob's baseline and strength-dimension parameters as can also be observed in Figure 1.


Finally, in Figure 5 (top graphs) we can observe the difference between the two (2) simulations in terms of the limiting probabilities π0, π1, and π2 in the case where Bob and Alice would never meet, Alice would be in red most of the time while Bob would be in green, yet these tendencies are changed by their interactions as can be clearly observed in Figure 5 (top left).


CONCLUSIONS AND FUTURE PERSPECTIVE

We have introduced the reader to a stochastic simulation model in order to describe the dynamics of two (2) agents who influence each other's HRV in a set of successive encounters during a certain horizon of time. We have learned via this modelling process about different aspects and ways these interactions may take place for people who embody polar tendency, stressed (red) and coherent (green) with different strength-dimension parameters.


So far we conclude, based on the present set of assumptions and simplifications, that people with a red tendency, who can be easily influenced, when interacting with people in green with strong dimension parameters with the power to influence others, will improve their condition. However, people with red tendency will also easily deteriorate when interacting with others embodying a strong red tendency. On the other hand, people with relatively strong green tendency, will compromise their own, when interacting with people with even relatively weak strength-dimension parameters. Whoever is stronger will attract the other towards his or her own tendency.


Even though our model was focused on HRV as a biomarker for psychophysiological coherence, other biomarkers could be used, such as, brain waves measured via EEG or stress hormone levels. Also, this model would allow us to simulate existential states as soft variables (Hayward et al., 2014), in order to model the dynamics of polar variables like love and hate or the level of inner peace (Zhuang et al., 2016) and stress, for example.


There are some technical considerations that could add value to modelling this process like incorporating Semi-Markovian or ARIMA models (Howard, 1971), (Macridakis et al., 1978) when necessary, since learning or developing a green tendency, for example, may introduce memory into the system dynamics, creating longer transit times for certain states, than the usual Markovian process transit times.


Finally, we foresee that this seminal model will serve as a stepping stone in modelling a multi-agent system where members of the collective learn how to fashion harmonious and coherent (green) societies (Davis & Schübeler, 2018), (Davis & Schübeler, 2019b), (Auroville, 2018).


ACKNOWLEDGEMENTS

We would like to acknowledge the team at The Embassy of Peace in Whitianga, New Zealand, for their invaluable participation. Particularly, Kali, Colin, Enya, Carey, Sarah, Shiloh, Matthew and Keryn.


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