was aimed at trying to formally capture what is consciousness in living beings [3, 7, 8] and the reader is referred to Tononi's papers for detailed motivations of the model.

We have considered the discrete version of the model [1, 2, 7]. The original papers describing the model are not always fully clear in their mathematical formulation and here we have given the first formal description of such a model where all steps are detailed presented.

In doing so we have provided a more general formulation of such a model, which is independent from the time chosen for the analysis and from the uniformity of the probability distribution at the initial time instant.

Finally, we have also given here the first formal proof that a system made up by independent parts has a value of integrated information equal to zero.

Acknowledgments. We would like to thank Luciano Gualà and Guido Proietti for useful and interesting discussions related to the work here described.

## References

[1] David Balduzzi and Giulio Tononi. Integrated information in discrete dynamical systems: Motivation and theoretical framework. *PLoS Computational Biology*, 4(6), 2008.

[2] David Balduzzi and Giulio Tononi. Qualia: The geometry of integrated information. PLoS Computational Biology, 5(8), 2009.

[3] Gerald M. Edelman and Giulio Tononi. *A universe of consciousness: how matter becomes imagination*. New York, NY: Basic Books, 1 edition, 2000.

[4] Ilya Shmulevich, Edward R. Dougherty, Seungchan Kim, and Wei Zhang. Probabilistic boolean networks: A
rule-based uncertainty model for gene regulatory networks. *Bioinformatics*, 18(2):261–274, 2002.

[5] Giulio Tononi. Information measures for conscious experience. *Archivi Italiani di Biologia*, 139(4):367–371, 2001.
[6] Giulio Tononi. An information integration theory of consciousness. *BMC Neuroscience*, 5(1):42, 2004.

[7] Giulio Tononi. Consciousness as integrated information: a provisional manifesto. *Biol. Bull.*, December 2008.

[8] Giulio Tononi and Gerald M. Edelman. Consciousness and complexity. *Science*, 282(5395):1846–1851, 1998.

[9] Giulio Tononi, Gerald M. Edelman, and Sporns Olaf. Complexity and coherency: integrating information in the brain. *Trends in Cognitive Sciences*, 2(11), 1998.

[10] Giulio Tononi and Olaf Sporns. Measuring information integration. *BMC Neuroscience*, 4, December 2003.

[11] Giulio Tononi, Olaf Sporns, and Gerald M. Edelman. A measure for brain complexity: relating functional segregation and integration in the nervous system. *Proceedings of the National Academy of Sciences of the* United States of America, 91(11):5033–5037, 1994.

[12] Giulio Tononi, Olaf Sporns, and Gerald M. Edelman. A complexity measure for selective matching of signals by the brain. *Proceedings of the National Academy of Sciences of the United States of America*, 93:3422, 1996.