The information theory of individuality by Krakauer, D., Bertschinger, N., Olbrich, E., Flack, J. C., & Ay, N. (2020)

Information theory, Life
(Krakauer et al. 2020)


This paper introduces an information theoretic definition of individuality for complex systems.

In a few words, the authors idea of individuality is based on the amount of information transmitted through time.

If the information transmitted forward in time is close to maximal, we take that as evidence for individuality.

Formally, a system \(\mathcal{S}\) is considered in interaction with an environment \(\mathcal{E}\). Two channels \(\phi: \mathcal{E} \times \mathcal{S}\rightarrow\mathcal{S}\) and \(\psi : \mathcal{S} \times \mathcal{E} \rightarrow \mathcal{E}\) corresponds to the transition probabilities for \(\mathcal{S}\) and \(\mathcal{E}\) respectively (\(\phi(e, s; s')\) is the probability to be in state \(s'\) at the next time step for current states \(s\) and \(e\) for \(\mathcal{S}\) and \(\mathcal{E}\)).

The predictability of the next state of the system is characterized in terms of entropy and mutual information by

\begin{align*} 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{align*}

Depending on which quantity dominates, different types of individuality can be defined

  • Colonial Individuality: \(A = I(S_{n+1} ; S_n|E_n)\)

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.

  • Organismal Individuality: \(A^* = I(S_{n+1} ; S_n)\)

Many organisms such as microbes share only a small amount of information with the environment in which hey 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 Determined Individuality: \(nC = I(S_{n+1} ; E_n|S_n)\)

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.

Another additional measure is defined, quantifying contribution of different terms in case of hybrid that have multiple components to their “individuality”. This is called Environmental Coding,

\[ NTIC=SI(S_{n+1};S_n, E_n)−CI(S_{n+1};S_n, E_n) \]

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.


This looks like a great way to study emergence of life and evolutionary processes in complex systems. I gives theoretical grounding into the fact that many complex systems look as if they behaved as individuals.

However, the main drawback seems to be in accurately computing mutual information for each of the terms. Also, the definition of the system \(\mathcal{S}\) and environment \(\mathcal{E}\) can be problematic when dealing with systems in which the distinction isn’t clear (such as Cellular automata).


Krakauer, David, Nils Bertschinger, Eckehard Olbrich, Jessica C. Flack, and Nihat Ay. 2020. “The Information Theory of Individuality.” Theory in Biosciences 139 (2):209–23.

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