The information theory of individuality

In particular, we assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (e,s;s') \ge 0$$\end{document}φ(e,s;s′)≥0 for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e,s,s'$$\end{document}e,s,s′, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{s'} \varphi (e,s;s') = 1$$\end{document}∑s′φ(e,s;s′)=1 for all e, s.

We can model the dynamics of the environment in the same way, using a Markov kernel \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi : {{\mathcal {S}}} \times {{\mathcal {E}}} \rightarrow {{\mathcal {E}}}$$\end{document}ψ:S×E→E, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (s,e;e')$$\end{document}ψ(s,e;e′) denotes the probability for the next state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e'$$\end{document}e′ of the environment given the current states e and s of the environment and the system, respectively. The kernels \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi$$\end{document}φ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi$$\end{document}ψ model the mechanisms that constitute the system–environment interaction.

If we start this interaction process by selecting a states s and e according to some probability distribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document}μ, we obtain a process \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(S_k,E_k)$$\end{document}(Sk,Ek), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k = 1,2,\ldots$$\end{document}k=1,2,…, in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {S}}} \times {{\mathcal {E}}}$$\end{document}S×E that satisfies\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&{\mathbb P}(S_1 = s_1, E_1 = e_1, S_2 = s_2, E_2 = e_2, \ldots , S_n \\&\quad = s_n, E_n = e_n) \\&\quad = \mu (s_1,e_1) \, \varphi (e_1,s_1;s_2) \psi (s_1,e_1;e_2) \, \ldots \, \\&\qquad \varphi (e_{n - 1} , s_{n - 1} ; s_n) \psi (s_{n - 1} , e_{n - 1} ; e_n), \qquad n = 1, 2, \ldots .

\end{aligned}$$\end{document}P(S1=s1,E1=e1,S2=s2,E2=e2,…,Sn=sn,En=en)=μ(s1,e1)φ(e1,s1;s2)ψ(s1,e1;e2)…φ(en-1,sn-1;sn)ψ(sn-1,en-1;en),n=1,2,….Clearly, we can recover the mechanisms from the distribution of the process \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(S_k, E_k)$$\end{document}(Sk,Ek), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k = 1,\ldots ,$$\end{document}k=1,…,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathbb P}(S_1&= {} s, E_1 = e) = \mu (s,e), \\ {\mathbb P}(S_k&= {} s' \, | \, E_{k - 1} = e, S_{k - 1} = s) = \varphi (e , s ; s' ), \\ {\mathbb P}(E_k &= {} e' \, | \, S_{k - 1} = s, E_{k - 1} = e) = \psi (s , e ; e' ). \end{aligned}$$\end{document}P(S1=s,E1=e)=μ(s,e),P(Sk=s′|Ek-1=e,Sk-1=s)=φ(e,s;s′),P(Ek=e′|Sk-1=s,Ek-1=e)=ψ(s,e;e′).We apply information-theoretic quantities, such as the mutual information, to variables of the process \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(S_k, E_k)$$\end{document}(Sk,Ek), thereby quantifying information flows between the system and the environment. The causal structure of the process, as shown in Fig. 1, implies a number of conditional independence statements.

For instance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{n + 1}$$\end{document}En+1 is conditionally independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_n, E_n$$\end{document}Sn,En given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{n-1}, E_{n-1}$$\end{document}Sn-1,En-1.

In the previous section, we set down the information-theoretic foundations for our formalism. Here, we discuss the additional mathematical properties required of the formalism if it is to capture the concept of individuality we developed in “A way forward” section.

We remind the reader our starting point is the assumption that biological individuality can usefully be understood as an “informational individual.” We further remind the reader this is not to be confused with Dawkin’s replicator, as we want to allow the possibility that replication is not a fundamental feature of individuality and be able to ask what role individuality plays in facilitating replication. What is fundamental in our view is the idea that information can be propagated forward through time, meaning that uncertainty is reduced over time. In this way, and returning to our opening remarks in “A way forward” section, we suggest individuality is a natural extension of the ideas of Boltzmann and Von Neumann, and as such has foundations in statistical mechanics and thermodynamics, which consider the conditions required for a persistently ordered states.

Defining properties and implications of the formalism The system environment decomposition Consider a dynamical set of quantifiable measurements that we coarse grain into components of a system and components of an environment. We seek a way of establishing whether this partition is justifiable, and whether the individuality concept is relevant. We wish to allow for a hierarchy of such partitions in order to capture biological examples such as organelles within cells, and cells within bodies within populations, where in each case the target entity and the environment assume a different identity. We retain those partitions that meet our information-theoretic inclusion criteria and then can ask which among the natural, intuitive categories of biology—e.g., cells, organelles, organisms, populations, etc., are recovered.Informational individuals In the pursuit of generality, we consider a discrete, stochastic process where the state of the system in the future is determined by some subset of states in the present.

If we arbitrarily divide these states into system and environment, we should like to be able to determine how the current system state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_n$$\end{document}Sn and the current state of the environment \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_n$$\end{document}En together are sufficient to determine the next system state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{n+1}$$\end{document}Sn+1. Formally, the predictability of the next state of the system is quantified via the mutual information: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I(S_n, E_n; \,S_{n+1}) = H(S_{n+1}) - H(S_{n+1} | S_n, E_n) . \end{aligned}$$\end{document}I(Sn,En;Sn+1)=H(Sn+1)-H(Sn+1|Sn,En).

This expression seeks to capture how much information at time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n+1$$\end{document}n+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{n+1}$$\end{document}Sn+1 comes from the system itself at a previous time step (or generation) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_n$$\end{document}Sn—the individual—versus from the environment at a previous time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_n$$\end{document}En. This mutual information can now be decomposed in two ways \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I(S_n, E_n; S_{n+1})&= {} I(S_{n+1}; S_n) + I(S_{n+1}; E_n | S_n) \\&= {} I(S_{n+1}; E_n) + I(S_{n+1}; S_n | E_n) \end{aligned}$$\end{document}I(Sn,En;Sn+1)=I(Sn+1;Sn)+I(Sn+1;En|Sn)=I(Sn+1;En)+I(Sn+1;Sn|En) Each decomposition can be interpreted as different allocation for distributing the observed past regularities between the system and environment.

Each of these will allow us to define different forms of individuality. aEndogenous determination Consider \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I(S_{n+1}; S_n) + I(S_{n+1}; E_n | S_n)$$\end{document}I(Sn+1;Sn)+I(Sn+1;En|Sn):Here, we measure the influence of the system state onto itself (at the next generation or time step). For a preferred interval of time, all observed dependencies between successive system states are attributed to the system.The quantity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I(S_{n+1}; S_n)$$\end{document}I(Sn+1;Sn) has been called autonomy in Krakauer and Zanotto (2006) and will be denoted as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^*$$\end{document}A∗ in the following. It should be high when the system is largely control of its environment.The influence of the environment, as measured by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I(S_{n+1}; E_n | S_n)$$\end{document}I(Sn+1;En|Sn), can be interpreted as new information for the system flowing from the environment into the system. When this information flow vanishes completely, a system can be said to be informationally closed.

So this quantity measures the degree to which the system is controlled by the environment nC. Note that closure does not require causal independence, it only states that all influences from the environment are predictable by the system.bEnvironmentally driven An alternative to endogenous determination is structure imposed largely through environmental gradients driving the system. In other words, the history of the system is not as consequential as the history of the environment that impose strong boundary conditions on the system. Consider \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I(S_{n+1}; E_n) + I(S_{n+1}; S_n | E_n)$$\end{document}I(Sn+1;En)+I(Sn+1;Sn|En):Here, the observed influences are attributed to the environment (as far as possible according to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I(S_{n+1}; E_n)$$\end{document}I(Sn+1;En)). Only the remaining influence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I(S_{n+1}; S_n |E_n)$$\end{document}I(Sn+1;Sn|En) is due to the system. This can be interpreted as an alternative concept of system autonomy (Bertschinger et al. 2008) and will be denoted as A in the following. It is valid under the assumption that all dependencies between the states of the system and the environment are attributed to the environment.

These properties allow us to identify three quantities, each corresponding to a type of individuality: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text{ Colonial } \text{ Individuality } \quad A&:=I(S_{n+1};S_n|E_n) \\ \text{ Organismal } \text{ Individuality } \quad A^{*}&:=I(S_{n+1};S_n) \\ \text{ Environmental } \text{ Determined } \text{ Individuality } \quad nC&:=I(S_{n+1};E_n|S_n) \end{aligned}$$\end{document}ColonialIndividualityA:=I(Sn+1;Sn|En)OrganismalIndividualityA∗:=I(Sn+1;Sn)EnvironmentalDeterminedIndividualitynC:=I(Sn+1;En|Sn) To rigorously formalize these different types of individuality, however, we need to consider them on a more fine-grained scale.

Using the chain rule for mutual information, we encounter an ambiguity in attributing influence to the environment or to the system. The partial information decomposition (Williams and Beer 2010; Bertschinger et al. 2013). allows us to resolve this ambiguity by introducing notions of unique, shared and complementary information.1

The mutual information between the future state of the system at time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n+1$$\end{document}n+1 and the joint state of system and environment at time n is decomposed into four terms:1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I(S_{n+1};S_n,E_n)&= \underbrace{SI(S_{n+1};S_n,E_n)}_{\text {shared}} + \underbrace{CI(S_{n+1}; S_n,E_n)}_{\text {complementary}} \\&\quad+ \underbrace{UI(S_{n+1};S_n\backslash E_n)}_{\text {unique }(S_n\text { wrt }E_n)} + \underbrace{UI(S_{n+1};E_n\backslash S_n)}_{\text {unique }(E_n\text { wrt } S_n)}.

\end{aligned}$$\end{document}I(Sn+1;Sn,En)=SI(Sn+1;Sn,En)⏟shared+CI(Sn+1;Sn,En)⏟complementary+UI(Sn+1;Sn\En)⏟unique(SnwrtEn)+UI(Sn+1;En\Sn)⏟unique(EnwrtSn).Those four terms appear in the pairwise mutual information and conditional mutual information that we obtained from the chain rule:2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I(S_{n+1};S_n)&= SI(S_{n+1};S_n,E_n) + UI(S_{n+1};S_n\backslash E_n) , \end{aligned}$$\end{document}I(Sn+1;Sn)=SI(Sn+1;Sn,En)+UI(Sn+1;Sn\En),3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I(S_{n+1};E_n|S_n)&= CI(S_{n+1};S_n,E_n) + UI(S_{n+1};E_n\backslash S_n) , \end{aligned}$$\end{document}I(Sn+1;En|Sn)=CI(Sn+1;Sn,En)+UI(Sn+1;En\Sn),4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I(S_{n+1};E_n)&= SI(S_{n+1};S_n,E_n) + UI(S_{n+1};E_n\backslash S_n) ,\end{aligned}$$\end{document}I(Sn+1;En)=SI(Sn+1;Sn,En)+UI(Sn+1;En\Sn),5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I(S_{n+1};S_n|E_n)&= CI(S_{n+1};S_n,E_n) + UI(S_{n+1};S_n\backslash E_n) , \end{aligned}$$\end{document}I(Sn+1;Sn|En)=CI(Sn+1;Sn,En)+UI(Sn+1;Sn\En),In our context the four terms have the following

meaning: aThe unique information from the system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$UI(S_{n+1};S_n \setminus E_n)$$\end{document}UI(Sn+1;Sn\En). This is information maintained by the system.bThe shared information between the system and environment \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SI(S_{n+1},S_n,E_n)$$\end{document}SI(Sn+1,Sn,En).cThe unique information from the environment \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$UI(S_{n+1};E_n \setminus S_n)$$\end{document}UI(Sn+1;En\Sn). This quantifies the influence of the environment on the system. (Information flow in the narrow sense).dThe complementary or synergistic information. Information that is only present in the interaction of systems and environment.It is important to emphasize that these decompositions are a means of supporting our formal intuition and do not correspond to a specification of the information-theoretic quantities. This choice remains disputed and several alternative proposals have been published. These are reviewed in a special issue of the journal Entropy (Lizier et al. 2018). Nevertheless, the measures that we derive fully accord with the conceptual decomposition.

With a good understanding of the implications of partial information decomposition in hand in hand, we can now rigorously define three forms of individuality and an additional measure quantifying contribution of each in the case of hybrid types. These measures are defined in terms of the information that is shared by system and environment (e.g., adaptive information), information that is unique to either the system or the environment (e.g., memory in each), and information that depends in some complicated way on both the system and the environment (e.g., regulatory information).Organismal Individuality\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^*$$\end{document}A∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A^{*}=SI(S_{n+1};S_n,E_n) + UI(S_{n+1};S_n\backslash E_n) \end{aligned}$$\end{document}A∗=SI(Sn+1;Sn,En)+UI(Sn+1;Sn\En) Organisms are well adapted when they share through adaptation or learning significant information with the environment in which they live. In addition, they contain a large amount of private information required for effective function.

By maximizing this measure, we are able to identify complex organisms in their environments.Colonial IndividualityA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A=CI(S_{n+1};S_n,E_n) + UI(S_{n+1};S_n\backslash E_n) \end{aligned}$$\end{document}A=CI(Sn+1;Sn,En)+UI(Sn+1;Sn\En) Many organisms such as microbes share only a small amount of information with the environment in which they live. They contain regulatory mechanisms that allow for adaptation through ongoing interaction between their biotic and abiotic environment. By maximizing this measure, we are able to identify “environmentally regulated aggregations,” which we call “colonial individuals.”Environmental determinationnC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} nC&= {} I(S_{n+1};E_n|S_n) = CI(S_{n+1};S_n,E_n) \\&\quad + UI(S_{n+1};E_n\backslash S_n) \end{aligned}$$\end{document}nC=I(Sn+1;En|Sn)=CI(Sn+1;Sn,En)+UI(Sn+1;En\Sn) This measure quantifies the degree of environmental determinism on the temporal evolution of an individual. When this measure is minimized an individual becomes completely insensitive to the environment—and hence is neither in the organismal or colonial form—and not in any real sense adaptive.

It represents the persistence of an environmental memory capable through interaction with the system of generating structure, such as temperature gradients in a fluid that produce vortices.Environmental Coding\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} NTIC=SI(S_{n+1};S_n,E_n) - CI(S_{n+1};S_n,E_n) \end{aligned}$$\end{document}NTIC=SI(Sn+1;Sn,En)-CI(Sn+1;Sn,En) The intuition behind this measure is to quantify the difference between a colonial and organismal measure of individuality. The difference is captured by the difference between shared information (e.g., adaptive information) and the interaction of individual and environment (e.g., regulatory information). One way to think about this is how much information can be encoded about the environment in the system innately (e.g., inherited information) versus how much information needs to be encoded through ongoing interaction. When the measure is large nature dominates nurture. As the measure declines, nurture begins to dominate nature.

To gain a better understanding of each of these measures, we work through a quantitative example.

We consider two binary units \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_n$$\end{document}En and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_n$$\end{document}Sn, with state sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{-1, + 1\}$$\end{document}{-1,+1}. Following the general structure introduced in sect. 2.1 and Fig.

1, these states are synchronously updated according to the following conditional distribution:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p(s_{n+1},e_{n+1} | s_n,e_n) \, = \, p_S(s_{n+1} | s_n, e_n) \cdot p_E(e_{n+1} | s_n,e_n), \end{aligned}$$\end{document}p(sn+1,en+1|sn,en)=pS(sn+1|sn,en)·pE(en+1|sn,en),where6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p_S(s_{n+1} | s_n, e_n)= {} \frac{1}{1 + \mathrm{e}^{- 2 s_{n+1} \left( \delta _S + \alpha _S s_n + \beta _S e_n + \gamma _S s_n e_n\right) }} \end{aligned}$$\end{document}pS(sn+1|sn,en)=11+e-2sn+1δS+αSsn+βSen+γSsnen7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p_E(e_{n+1} | s_n, e_n)= \frac{1}{1 + \mathrm{e}^{- 2 e_{n+1} \left( \delta _E + \alpha _E e_n + \beta _E s_n + \gamma _E s_n e_n\right) }}.

\end{aligned}$$\end{document}pE(en+1|sn,en)=11+e-2en+1δE+αEen+βEsn+γEsnen.Evaluating the individual conditional distributions, we obtain\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p_S(+1 | +1, +1)&= \frac{1}{1 + \mathrm{e}^{- 2 \left( \delta _S + \alpha _S + \beta _S + \gamma _S \right) }} =: a_S \\ p_S(+ 1 | -1, +1)&= \frac{1}{1 + \mathrm{e}^{- 2 \left( \delta _S - \alpha _S + \beta _S - \gamma _S \right) }} =: b_S \\ p_S(+ 1 | +1,-1)&= \frac{1}{1 + \mathrm{e}^{- 2 \left( \delta _S + \alpha _S - \beta _S - \gamma _S \right) }} =: c_S \\ p_S(+ 1 | -1,-1)&= \frac{1}{1 + \mathrm{e}^{- 2 \left( \delta _S - \alpha _S - \beta _S + \gamma _S \right) }} =: d_S \end{aligned}$$\end{document}pS(+1|+1,+1)=11+e-2δS+αS+βS+γS=:aSpS(+1|-1,+1)=11+e-2δS-αS+βS-γS=:bSpS(+1|+1,-1)=11+e-2δS+αS-βS-γS=:cSpS(+1|-1,-1)=11+e-2δS-αS-βS+γS=:dSand correspondingly\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p_E(+1 | +1, +1)&= {} \frac{1}{1 + \mathrm{e}^{- 2 \left( \delta _E + \alpha _E + \beta _E + \gamma _E \right) }} =: a_E \\ p_E(+ 1 | -1, +1)&= {} \frac{1}{1 + \mathrm{e}^{- 2 \left( \delta _E + \alpha _E - \beta _E - \gamma _E \right) }} =: b_E \\ p_E(+ 1 | +1,-1)&= {} \frac{1}{1 + \mathrm{e}^{- 2 \left( \delta _E - \alpha _E + \beta _E - \gamma _E \right) }} =: c_E \\ p_E(+ 1 | -1,-1)&= {} \frac{1}{1 + \mathrm{e}^{- 2 \left( \delta _E - \alpha _E - \beta _E + \gamma _E \right) }} =: d_E \end{aligned}$$\end{document}pE(+1|+1,+1)=11+e-2δE+αE+βE+γE=:aEpE(+1|-1,+1)=11+e-2δE+αE-βE-γE=:bEpE(+1|+1,-1)=11

+e-2δE-αE+βE-γE=:cEpE(+1|-1,-1)=11+e-2δE-αE-βE+γE=:dEFinally, this yields the following stochastic matrix with entries \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(s_{n+1},e_{n+1} | s_n,e_n)$$\end{document}p(sn+1,en+1|sn,en):\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{array}{ c || c | c | c | c |} &{} (+ 1, + 1) &{} (- 1, + 1) &{} (+ 1, -1) &{} (-1 , -1) \\ \hline \hline (+ 1, + 1) &{} a_S a_E &{} (1 - a_S) a_E &{} a_S (1 - a_E) &{} (1 - a_S) (1 - a_E) \\ (- 1, + 1) &{} b_S b_E &{} (1 - b_S) b_E &{} b_S (1 - b_E) &{} (1 - b_S) (1 - b_E) \\ (+ 1, - 1) &{} c_S c_E &{} (1 - c_S) c_E &{} c_S (1 - c_E) &{} (1 - c_S) (1 - c_E) \\ (- 1, - 1) &{} d_S d_E &{} (1 - d_S) d_E &{} d_S (1 - d_E) &{} (1 - d_S) (1 - d_E) \\ \hline \end{array} \end{aligned}$$\end{document}(+1,+1)(-1,+1)(+1,-1)(-1,-1)(+1,+1)aSaE(1-aS)aEaS(1-aE)(1-aS)(1-aE)(-1,+1)bSbE(1-bS)bEbS(1-bE)(1-bS)(1-bE)(+1,-1)cScE(1-cS)cEcS(1-cE)(1-cS)(1-cE)(-1,-1)dSdE(1-dS)dEdS(1-dE)(1-dS)(1-dE)Fig.

2Mutual information between two time steps (Total_MI), Entropy of the system (H_sys), colonial (A) and organismal (A_star) individuality, and environmental determination (nC) for different values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _S$$\end{document}αS,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _S$$\end{document}βS, and for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _S$$\end{document}γS (subscript “S” omitted in the figure) with a random environment \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _E=\beta _E=\gamma _E=0$$\end{document}αE=βE=γE=0Fig.

3Mutual information between two time steps (Total_MI), Entropy of the system (H_sys), colonial (A) and organismal (A_star) individuality, and environmental determination (nC) for different values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _S$$\end{document}αS,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _S$$\end{document}βS, and for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _S$$\end{document}γS with a correlated environment \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _E=2 \quad \beta _E=\gamma _E=0$$\end{document}αE=2βE=γE=0

We apply each individuality measure to this stochastic process. The results of this analysis are shown in Fig. 2 for a random environment and in Fig. 3 for an environment with memory. The panel sweeps through three coupling parameters for the systems state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_{n+1}$$\end{document}sn+1: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _S$$\end{document}αS—the coupling parameter of the system state to its previous state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_n$$\end{document}sn, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _S$$\end{document}βS—the coupling parameter to the environment, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _S$$\end{document}γS the coupling parameter mediating the combined influence of the previous system and environmental states.

When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _S=0$$\end{document}γS=0, we are not imposing any higher-order correlations on the time series.

When the value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _S=0$$\end{document}γS=0, we detect colonial individuals as well as organismal individuals most readily at high values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _S$$\end{document}αS and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _S$$\end{document}βS. When there are no higher-order interactions between system and environment then these two types of individual become indistinguishable in this parameter region and represent unique information in the system state. Both forms of individuality become more visible as more information is transmitted into the future.