# American Institute of Mathematical Sciences

October  2020, 40(10): 5765-5793. doi: 10.3934/dcds.2020245

## On the effects of firing memory in the dynamics of conjunctive networks

 1 Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Diagonal las Torres 2650, Peñalolen, Santiago, Chile, and, Unconventional Computing Laboratory, University of the West of England, Bristol, UK 2 Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Diagonal las Torres 2650, Peñalolen, Santiago, Chile 3 Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Beauchef 851, Torre Norte, Piso 5, Santiago, Chile, and, Aix Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, France

* Corresponding author: Pedro Montealegre

Some of the results of this paper have been presented in the 25th International Workshop on Cellular Automata and Discrete Complex Systems AUTOMATA 2019.

Received  July 2019 Revised  February 2020 Published  June 2020

A boolean network is a map $F:\{0,1\}^n \to \{0,1\}^n$ that defines a discrete dynamical system by the subsequent iterations of $F$. Nevertheless, it is thought that this definition is not always reliable in the context of applications, especially in biology. Concerning this issue, models based in the concept of adding asynchronicity to the dynamics were propose. Particularly, we are interested in a approach based in the concept of delay. We focus in a specific type of delay called firing memory and it effects in the dynamics of symmetric (non-directed) conjunctive networks. We find, in the caseis in which the implementation of the delay is not uniform, that all the complexity of the dynamics is somehow encapsulated in the component in which the delay has effect. Thus, we show, in the homogeneous case, that it is possible to exhibit attractors of non-polynomial period. In addition, we study the prediction problem consisting in, given an initial condition, determinate if a fixed coordinate will eventually change its state. We find again that in the non-homogeneous case all the complexity is determined by the component that is affected by the delay and we conclude in the homogeneous case that this problem is PSPACE-complete.

Citation: Eric Goles, Pedro Montealegre, Martín Ríos-Wilson. On the effects of firing memory in the dynamics of conjunctive networks. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5765-5793. doi: 10.3934/dcds.2020245
##### References:
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Montealegre, Computational complexity of threshold automata networks under different updating schemes, Theoret. Comput. Sci., 559 (2014), 3-19.  doi: 10.1016/j.tcs.2014.09.010. [11] E. Goles, P. Montealegre, V. Salo and I. Törmä, PSPACE-completeness of majority automata networks, Theoret. Comput. Sci., 609 (2016), 118-128.  doi: 10.1016/j.tcs.2015.09.014. [12] E. Goles and M. Noual, Disjunctive networks and update schedules, Adv. in Appl. Math., 48 (2012), 646-662.  doi: 10.1016/j.aam.2011.11.009. [13] E. Goles and L. Salinas, Comparison between parallel and serial dynamics of Boolean networks, Theoret. Comput. Sci., 396 (2008), 247-253.  doi: 10.1016/j.tcs.2007.09.008. [14] E. Goles-Chacc, F. Fogelman-Soulié and D. Pellegrin, Decreasing energy functions as a tool for studying threshold networks, Discrete Appl. Math., 12 (1985), 261-277.  doi: 10.1016/0166-218X(85)90029-0. [15] A. Graudenzi and R. Serra, A new model of genetic network: The Gene Protein Boolean network, Artificial Life and Evolutionary Computation, World Sci. Publ., Hackensack, NJ, (2010), 283-291. doi: 10.1142/9789814287456_0025. [16] A. Graudenzi, R. Serra, M. Villani, A. Colacci and S. A. Kauffman, Robustness analysis of a Boolean model of gene regulatory network with memory, J. Comput. Biol., 18 (2011), 559-577.  doi: 10.1089/cmb.2010.0224. [17] A. Graudenzi, R. Serra, M. Villani, C. Damiani, A. Colacci and S. A. Kauffman, Dynamical properties of a Boolean model of gene regulatory network with memory, J. Comput. Biol., 18 (2011), 1291-1303.  doi: 10.1089/cmb.2010.0069. [18] B. M. Gummow, J. O. Scheys, V. R. Cancelli and G. D. Hammer, Reciprocal regulation of a glucocorticoid receptor-steroidogenic factor-1 transcription complex on the Dax-1 promoter by glucocorticoids and adrenocorticotropic hormone in the adrenal cortex, Molecular Endocrinology, 20 (2006), 2711-2723. [19] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth edition. The Clarendon Press, Oxford University Press, New York, 1979. [20] B. Host, A. Maass and S. Martínez, Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules, Discrete Contin. Dyn. Syst., 9 (2003), 1423-1446.  doi: 10.3934/dcds.2003.9.1423. [21] A. S. Jarrah, R. Laubenbacher and A. Veliz-Cuba, The dynamics of conjunctive and disjunctive Boolean network models, Bull. Math. Biol., 72 (2010), 1425-1447.  doi: 10.1007/s11538-010-9501-z. [22] S. Kauffman, The large scale structure and dynamics of gene control circuits: An ensemble approach, Journal of Theoretical Biology, 44 (1974), 167-190. [23] S. A. Kauffman, Metabolic stability and epigenesis in randomly constructed genetic nets, Journal of Theoretical Biology, 22 (1969), 437-467.  doi: 10.1016/0022-5193(69)90015-0. [24] M. A. Kiwi, R. Ndoundam, M. Tchuente and E. Goles, No polynomial bound for the period of the parallel chip firing game on graphs, Theoret. Comput. Sci., 136 (1994), 527-532.  doi: 10.1016/0304-3975(94)00131-2. [25] D. H. Nguyen and P. D'haeseleer, Deciphering principles of transcription regulation in eukaryotic genomes, Molecular Systems Biology, 2 (2006). doi: 10.1038/msb4100054. [26] É. Remy, P. Ruet and D. Thieffry, Positive or negative regulatory circuit inference from multilevel dynamics, Notes Control Inf. Sci., 341 (2006), 263-270. [27] F. Ren and J. Cao, Asymptotic and robust stability of genetic regulatory networks with time-varying delays, Neurocomputing, 71 (2008), 834-842.  doi: 10.1016/j.neucom.2007.03.011. [28] T. Ribeiro, M. Magnin, K. Inoue and C. Sakama, Learning delayed influences of biological systems, Frontiers in Bioengineering and Biotechnology, 2 (2015), 81. doi: 10.3389/fbioe.2014.00081. [29] F. Robert, Discrete Iterations: A Metric Study, Springer Series in Computational Mathematics, 6. Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61607-5. [30] M. Sobottka, Right-permutative cellular automata on topological Markov chains, Discrete Contin. Dyn. Syst., 20 (2008), 1095-1109.  doi: 10.3934/dcds.2008.20.1095. [31] T. K. Subrahmonian Moothathu, Homogeneity of surjective cellular automata, Discrete Contin. Dyn. Syst., 13 (2005), 195-202.  doi: 10.3934/dcds.2005.13.195. [32] R. Thomas, Boolean formalization of genetic control circuits, Journal of Theoretical Biology, 42 (1973), 563-585. [33] R. Thomas, Regulatory networks seen as asynchronous automata: A Logical description, Journal of Theoretical Biology, 153 (1991), 1-23.  doi: 10.1016/S0022-5193(05)80350-9. [34] R. Thomas, D. Thieffry and M. Kaufman, Dynamical behaviour of biological regulatory networks-i. biological role of feedback loops and practical use of the concept of the loop-characteristic state, Bulletin of Mathematical Biology, 57 (1995), 247-276.

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##### References:
 [1] J. Ahmad, O. Roux, G. Bernot, J.-P. Comet and A. Richard, Analysing formal models of genetic regulatory networks with delays, International Journal of Bioinformatics Research and Applications, 4 (2008), 240-262.  doi: 10.1504/IJBRA.2008.019573. [2] J. Aracena, E. Goles, A. Moreira and L. Salinas, On the robustness of update schedules in Boolean networks, Biosystems, 97 (2009), 1-8.  doi: 10.1016/j.biosystems.2009.03.006. [3] J. Aracena, A. Richard and L. Salinas, Fixed points in conjunctive networks and maximal independent sets in graph contractions, J. Comput. System Sci., 88 (2017), 145-163.  doi: 10.1016/j.jcss.2017.03.016. [4] S. Arora and B. Barak, Computational Complexity: A Modern Approach, Cambridge University Press, Cambridge, 2009.  doi: 10.1017/CBO9780511804090. [5] G. Bernot, J.-P. Comet, A. Richard and J. Guespin, Application of formal methods to biological regulatory networks: Extending Thomas' asynchronous logical approach with temporal logic, J. Theoret. Biol., 229 (2004), 339-347.  doi: 10.1016/j.jtbi.2004.04.003. [6] J. Demongeot, A. Elena and S. Sené, Robustness in regulatory networks: A multi-disciplinary approach, Acta Biotheoretica, 56 (2008), 27-49.  doi: 10.1007/s10441-008-9029-x. [7] J. Fromentin, D. Eveillard and O. Roux, Hybrid modeling of biological networks: Mixing temporal and qualitative biological properties, BMC Systems Biology, 4 (2010), 79. doi: 10.1186/1752-0509-4-79. [8] Z. Gao, X. Chen and T. BaÅŸar, Controllability of conjunctive Boolean networks with application to gene regulation, IEEE Trans. Control Netw. Syst., 5 (2018), 770-781.  doi: 10.1109/TCNS.2017.2746345. [9] E. Goles, F. Lobos, G. A. Ruz and S. Sené, Attractor landscapes in Boolean networks with firing memory: A theoretical study applied to genetic networks, Nat. Comput., 19 (2020), 295-319.  doi: 10.1007/s11047-020-09789-0. [10] E. Goles and P. Montealegre, Computational complexity of threshold automata networks under different updating schemes, Theoret. Comput. Sci., 559 (2014), 3-19.  doi: 10.1016/j.tcs.2014.09.010. [11] E. Goles, P. Montealegre, V. Salo and I. Törmä, PSPACE-completeness of majority automata networks, Theoret. Comput. Sci., 609 (2016), 118-128.  doi: 10.1016/j.tcs.2015.09.014. [12] E. Goles and M. Noual, Disjunctive networks and update schedules, Adv. in Appl. Math., 48 (2012), 646-662.  doi: 10.1016/j.aam.2011.11.009. [13] E. Goles and L. Salinas, Comparison between parallel and serial dynamics of Boolean networks, Theoret. Comput. Sci., 396 (2008), 247-253.  doi: 10.1016/j.tcs.2007.09.008. [14] E. Goles-Chacc, F. Fogelman-Soulié and D. Pellegrin, Decreasing energy functions as a tool for studying threshold networks, Discrete Appl. Math., 12 (1985), 261-277.  doi: 10.1016/0166-218X(85)90029-0. [15] A. Graudenzi and R. Serra, A new model of genetic network: The Gene Protein Boolean network, Artificial Life and Evolutionary Computation, World Sci. Publ., Hackensack, NJ, (2010), 283-291. doi: 10.1142/9789814287456_0025. [16] A. Graudenzi, R. Serra, M. Villani, A. Colacci and S. A. Kauffman, Robustness analysis of a Boolean model of gene regulatory network with memory, J. Comput. Biol., 18 (2011), 559-577.  doi: 10.1089/cmb.2010.0224. [17] A. Graudenzi, R. Serra, M. Villani, C. Damiani, A. Colacci and S. A. Kauffman, Dynamical properties of a Boolean model of gene regulatory network with memory, J. Comput. Biol., 18 (2011), 1291-1303.  doi: 10.1089/cmb.2010.0069. [18] B. M. Gummow, J. O. Scheys, V. R. Cancelli and G. D. Hammer, Reciprocal regulation of a glucocorticoid receptor-steroidogenic factor-1 transcription complex on the Dax-1 promoter by glucocorticoids and adrenocorticotropic hormone in the adrenal cortex, Molecular Endocrinology, 20 (2006), 2711-2723. [19] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth edition. The Clarendon Press, Oxford University Press, New York, 1979. [20] B. Host, A. Maass and S. Martínez, Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules, Discrete Contin. Dyn. Syst., 9 (2003), 1423-1446.  doi: 10.3934/dcds.2003.9.1423. [21] A. S. Jarrah, R. Laubenbacher and A. Veliz-Cuba, The dynamics of conjunctive and disjunctive Boolean network models, Bull. Math. Biol., 72 (2010), 1425-1447.  doi: 10.1007/s11538-010-9501-z. [22] S. Kauffman, The large scale structure and dynamics of gene control circuits: An ensemble approach, Journal of Theoretical Biology, 44 (1974), 167-190. [23] S. A. Kauffman, Metabolic stability and epigenesis in randomly constructed genetic nets, Journal of Theoretical Biology, 22 (1969), 437-467.  doi: 10.1016/0022-5193(69)90015-0. [24] M. A. Kiwi, R. Ndoundam, M. Tchuente and E. Goles, No polynomial bound for the period of the parallel chip firing game on graphs, Theoret. Comput. Sci., 136 (1994), 527-532.  doi: 10.1016/0304-3975(94)00131-2. [25] D. H. Nguyen and P. D'haeseleer, Deciphering principles of transcription regulation in eukaryotic genomes, Molecular Systems Biology, 2 (2006). doi: 10.1038/msb4100054. [26] É. Remy, P. Ruet and D. Thieffry, Positive or negative regulatory circuit inference from multilevel dynamics, Notes Control Inf. Sci., 341 (2006), 263-270. [27] F. Ren and J. Cao, Asymptotic and robust stability of genetic regulatory networks with time-varying delays, Neurocomputing, 71 (2008), 834-842.  doi: 10.1016/j.neucom.2007.03.011. [28] T. Ribeiro, M. Magnin, K. Inoue and C. Sakama, Learning delayed influences of biological systems, Frontiers in Bioengineering and Biotechnology, 2 (2015), 81. doi: 10.3389/fbioe.2014.00081. [29] F. Robert, Discrete Iterations: A Metric Study, Springer Series in Computational Mathematics, 6. Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61607-5. [30] M. Sobottka, Right-permutative cellular automata on topological Markov chains, Discrete Contin. Dyn. Syst., 20 (2008), 1095-1109.  doi: 10.3934/dcds.2008.20.1095. [31] T. K. Subrahmonian Moothathu, Homogeneity of surjective cellular automata, Discrete Contin. Dyn. Syst., 13 (2005), 195-202.  doi: 10.3934/dcds.2005.13.195. [32] R. Thomas, Boolean formalization of genetic control circuits, Journal of Theoretical Biology, 42 (1973), 563-585. [33] R. Thomas, Regulatory networks seen as asynchronous automata: A Logical description, Journal of Theoretical Biology, 153 (1991), 1-23.  doi: 10.1016/S0022-5193(05)80350-9. [34] R. Thomas, D. Thieffry and M. Kaufman, Dynamical behaviour of biological regulatory networks-i. biological role of feedback loops and practical use of the concept of the loop-characteristic state, Bulletin of Mathematical Biology, 57 (1995), 247-276.
Symmetric conjunctive network ($F(x_0,x_1,x_2) = (x_1\land x_2, x_0\land x_2, x_0 \land x_1)$) dynamics in three vertices
Symmetric AND ($F(x_0,x_1,x_2) = (x_1\land x_2, x_0\land x_2, x_0 \land x_1)$) dynamics with firing memory $dt_i = 2, \text{ } i = 0,1,2$ in three vertices
Structure of a network $G^{\tau,k}$ in which the connected component $D$ has maximum delay $\tau \geq 2$ in every node and the connected component $G$ has maximum delay $\tau = 1$ in every node. Additionally, the subset $\partial$ contains every node in $D$ whose have a neighbour in $G$. (a).- The connected component $G$ has an arbiratry topology. (b).- The connected component $G$ is a bipartite graph. As it is shown in Proposition 1, the structure of $G$ may condition the existance of different type of attractors for a conjunctive network
A scheme of the dynamics of a node $i \in \partial$ that is in state 0 in some time step t. Note that before the sequence of states Ï„; Ï„ âˆ' 1;...; 0 we donâ€™t know the state of i and we note it as? in order to illustrate this situation. As a consequence of the fact that the maximum delay Ï„ is even together with the fact that two nodes in state 0 can not be neighbours we have that there are no attractor different from the fixed points $\overrightarrow 1$ and $\overrightarrow 0$.
(a) A network $G^{\tau,k}$ with maximum delay $\tau = 3$ and $10$ nodes exhibiting an attractor of period $8$. This network is defined by connecting the network $D$ (squares and non-colored circles) to a component with maximum delay $\tau = 1$ in every vertex (gray nodes). The nodes $1$ and $2$ (in grey) are the only nodes in $G^{\tau,k}$ with maximum delay $\tau = 1$. In addition, nodes represented by squares and non-coloured circles have maximum delay $\tau = 3$. Finally, nodes represented by squares are the nodes in the interface between the component $D$ and $G$ (we called this set $\partial$ in the latter section), and nodes represented by non-coloured circles are the ones that have no neighbours in $G$. (b) Configuration of an attractor with period $8$ defined over the network showed in (a)
Attractors with period $p = \tau +1$ for $\tau = 2$ and $\tau = 3$
Structure of the $j$-th block used in Proposition 5 to define a conjunctive network with firing memory and maximum delay values $dt_i = \tau$ for every node $i$ that admits attractors with period $k(\tau+1)$. Every circle in the figure represents clock $C(B_j)_{r,l}$ associated to a node $r$ represented by a square. This gadget has $\tau + 1$ nodes and every node has $\tau -1$ clocks
Initial condition for the j-th block, j â‰¥ 2 used in Proposition 5 to define an attractor with period k(Ï„ + 1): For the first block we just define the state of the first node to 0 instead of Ï„.
Interaction graph G associated to a conjunctive network with firing memory and maximum delay dti = Ï„ for all $i \in V\left( G \right)$, defined in Proposition 5 that admits attractors with length k(Ï„ + 1).
Structure of a block used in Proposition 5 to define a conjunctive network with firing memory and maximum delay values $dt_i = \tau$ for every node $i$ that admits attractors with period $k(\tau+1)$. Every circle in the figure represents clock associated to a node $r$ represented by a square. This gadget has $\tau + 1$ nodes and every node has $\tau -1$ clocks
Interaction graph $G$ associated to a conjunctive network with firing memory and maximum delay values $dt_i = \tau$ for every node $i \in V(G)$ that admits attractors with non polynomial period. Every component defines a local dynamics with period $(\tau + 1) p_i$. Initial condition is defined verifying that there are no connected nodes in $0$. Global period of the network is given by the product of prime numbers $p_i$.
Interaction graph associated to a conjunctive network with firing memory and maximum delay vector $dt_i = 2$ for every node $i$ that simulates an iterated monotone boolean circuit $C$. Layers are made up by AND or OR gates exclusively, using the gadgets shown in Figure 13 and Figure 14 are alternately ordered
Gadget of AND gates used in the graph shown in Figure 12. Signals are transmitted and coded based on the block gadget
Gadget of OR gates used in the graph shown in Figure 12. Signals are transmitted and coded based on the block gadget
Iterations of the AND gate gadget. A $0$ and a $1$ are computed as inputs. After seven steps the information is transmitted and initial condition is recovered.
Iterations of the OR gate gadget. A $0$ and a $1$ are computed as inputs. After three steps the information is transmitted and the initial condition is recovered.
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