- Email: [email protected]
A simple Hopfield-like cellular network model of plant intelligence
R. Banerjee & B.K. Chakrabarti (Eds.) Progress in Brain Research, Vol. 168 ISSN 0079-6123 Copyright r 2008 Elsevier B.V. All rights reserved
CHAPTER 14
A simple Hopfield-like cellular network model of plant intelligence Jun-ichi Inoue Complex Systems Engineering, Graduate School of Information Science and Technology, Hokkaido University, N14-W9, Kita-Ku, Sapporo 060-0814, Japan
Abstract: We introduce a simple Hopfield-like cellular-network model to explain a kind of ‘‘intelligence’’ in plants (Trewavas, 2002), especially, the capacity of plants to act as a memory device. Following earlier observations by Indian scientist J.C. Bose (1923), we regard the plant as a network in which each of the elements is connected via negative interactions. We investigate properties of the model by statistical mechanics. Keywords: plant intelligence; Hopfield model; associative memories; statistical-mechanical analysis; storage capacity; phase transition
the brain, is also based on parallel and distributed computation. Therefore, similarities between neural network models of brains and the plant networks should be discussed. Although the behavior of the dodder coil we mentioned above is due to the emergence of intelligence as a macroscopic function, it is important for us to investigate its rationale on a microscopic scale. Over 80 years ago, J.C. Bose (1923) detected electrical signaling between plant cells. Since then, following his experiments, many examples of cross-talk between biochemical signaling pathways in plants have been discovered. Especially, a Boolean representation of networks of signaling pathways is possible in terms of logical gates like AND, OR and XOR etc. These Boolean descriptions make it possible to draw analogies between plant networks and neural network models. Recently, Brueggemann et al. (1998) found the plant vacuolar membrane current–voltage characteristic to be equivalent to that of a Zenner diode. Inspired by their work, Chakrabarti and Dutta
Introduction Since pioneering studies by Indian scientist J.C. Bose (1923), plants have been regarded as a kind of network which are capable of intelligent responses to environmental stimuli. For example, the dodder coil, which is a plastic plant, explores a new host tree within hours subsequent to initial contact (Trewavas, 2002). This sort of behavior might be regarded as plant intelligence. If that is the case, does the plant compute, learn or memorize various spatial and temporal patterns in different environments similar to a computer or human brain (Ball, 2004)? Recently, Peak et al. (2004) pointed out that the plants may regulate their uptake and loss of gases by a distributed computation. As is well known, the ability of neural networks, which is a mathematical model of
Corresponding author. Tel.: +81-11-706-7225; Fax: +81-11-706-7391; E-mail: [email protected]
DOI: 10.1016/S0079-6123(07)68014-5
169
170
(2003) utilized the threshold behavior of plant cell membranes to develop or model gates for performing simple logical operations. They found that the plant network connections are all positive (excitatory by means of neuronal states) or all negative (inhibitory), compared to the randomly positive–negative distributed synaptic connections in real brains. As a result, the plant network does not involve any frustration in their computational capabilities and might lack the distributed parallel computational ability as in the case of associative memories. Even if we observe a single plant cell and provide a mathematical model for the single cell, we will not understand the macroscopic behavior of the plant giving rise to emergent properties like learning, memory and recognition (Genoud and Metraux, 1999; Bose and Karmakar, 2003). With this fact in mind, we investigate one of such collective behavior, namely, associative memories of plant networks, in this chapter. In other words, we try to answer the question ‘‘Do plant networks act as memory devices?’’ Especially, following the analysis by Chakrabarti and Dutta (2003), we investigate the equilibrium properties of the Hopfield model in which both ferromagnetic retrieval and anti-ferromagnetic terms co-exist. The strength of the anti-ferromagnetic order is controlled by a single parameter l. Within the replica symmetric calculation of statistical physics (Hertz et al., 1991; Nishimori, 2001), we obtain phase diagrams of the system. The l-dependence of the optimal loading rate ac, which is defined as the ratio of the maximum number of embedded patterns to the number of neurons in the brain, namely, Pmax/N, at T=0 is discussed. This chapter is organized as follows. In the next section, we introduce several experiments and observations related to the current–voltage characteristics of the plant cell membrane. In section ‘‘The Plant Intelligence Model’’, we model the plant possessing such properties by using a Hopfield-like network model in which both ferromagnetic-retrieval and anti-ferromagnetic terms exist. In section ‘‘Replica Symmetric Analysis’’, we analyze the model with the assistance of the replica method. In this section, we investigate to what extent the ferromagnetic retrieval order
remains against the anti-ferromagnetic disturbance. We also investigate the result of the ferromagnetic disturbance. All the results are summarized in the final section.
The I–V characteristics of cell membranes In this section, we briefly mention several results concerning properties of the plant units (cells), namely, current (I)–voltage (V) characteristics of their cell membrane. In Fig. 1, we show the typical non-linear I–V characteristics of cell membranes acting as logical gates. From this figure, we find that the I–V characteristics are equivalent to those of the Zenner diode. Namely, some threshold vT exists crossing which, the direction of the current changes. By assuming that the current has two directions, that is, 71, and the voltage is determined by the weighted contributions, Chakrabarti and Dutta (2003) constructed a mathematical plant cell as a non-linear unit. From the viewpoint of input– output logical units like perceptrons for neural networks, the output of the i-th unit Oi is given by ! N X wij I j Oi ¼ Y j¼1
where strength of each connection wij is all positive or negative, while in the Hopfield model it is given as 7 randomly distributed weight matrix in terms
1.5
0.5
5 mM
0 0
C2 vT(2) v
0
40 80 V (mV)
−0.5 −1
C1
1 mM
1
−60−40
l
l (A/m2)
vT(1) (a)
(b)
−1.5
Fig. 1. The non-linear I–V characteristics of cell membranes.
171
of the Hebbian rule wij ¼
P 1X xm xm N m¼1 i j
xmi 2 f1; 1g
From these experimental results and simple observations, we now ask a natural question, which is, could the plants act as memory devices similar to a real brain? Obviously, in the above definition of a single unit in the plant, there is no frustration as in animal brains. Thus, this paper attempts to elucidate, as to what extent constraints on the sign of the weight matrix influences the ability in plants to retrieve patterns as associative memories. For this purpose, we introduce a simple plant intelligence model based on a Hopfield-like model in which ferromagnetic retrieval and antiferromagnetic ordered phases co-exist. In the next section, we explain its details.
investigate the l-dependence of the system, namely, to study the l-dependence of the optimal loading rate ac(l) at T = 0 by using the technique of statistical mechanics for spin glasses.
Replica symmetric analysis In order to evaluate macroscopic properties of the system, we first evaluate the averaged free energy ½log Z~x ¼ ½log trS~ expðbHÞ~x where ½ ~x denotes the quenched average over the extensive P=aN patterns. To carry out this average and spin trace trS~ expðbHÞ, we use the replica method by using the relation ½log Z~x ¼ lim
n!0
The plant intelligence model We start from the following Hamiltonian ! P X 1X H¼ l xmi xmj Si S j ij N m¼1 ¼ H AF þ H FR where we defined the following two parts of the total Hamiltonian H AF ¼
lX Si Sj N ij
P 1 XX HF ¼ xm xm S i S j N ij m¼1 i j m In these expressions, the vector ~ x ¼ ðxm1 ; . . . ; xmN Þ is ~ ¼ ðS 1 ; :::; S N Þ stands m-th embedded pattern and S for neuronal states. A single parameter l determines the strength of the anti-ferromagnetic order, that is to say, in the limit of l-N, the system is completely determined by the energy function HAF. On the other hand, in the limit of l-0, the system becomes identical to the conventional Hopfield model. The purpose of this chapter is to
½Z n ~x 1 n
After standard algebra, we obtain the patternaveraged replicated partition function under the replica symmetric approximation as follows ½log Z~x ¼ extr expðnNFðm; q; M; rÞÞ m;q;M;r
with Fðm; q; M; rÞ ¼
b 2 bl 2 ab2 r M m þ ð1 qÞ 2 2 2 a bq þ log½1 bð1 qÞ 2 1 bð1 qÞ Z 1 pffiffiffiffiffi log Dz log 2 cosh bðlm þ arz þ MÞ 1
pffiffiffiffiffiffi where we define Dz ¼ dz expðz2 =2Þ= 2p. We should keep in mind that physical meanings of m and M are magnetization of the system, overlap ~ and a specific between the neuronal state S 1 recalling pattern ~ x among p=aN embedded patterns, respectively. The order parameter q stands for spin glass order parameters. In the next section, we evaluate the saddle point of this free energy density F and draw phase diagrams to specify the pattern retrieval properties of the system.
172 0.14
Phase diagrams
0.12
Z
pffiffiffiffiffi Dz tanh b½ð1 lÞM þ z ar ¼ m
1 1
0.06
0.02 0
pffiffiffiffiffi Dztanh b ð1 lÞM þ z ar
0
0.2
1
0.4
0.6
0.8
1
λ
2
q¼
r¼
0.08
0.04 1
M¼ Z
T=0 0.1 αc
In this section, we investigate the phase diagram of the system by solving the saddle point equations. By taking the derivatives of the free energy density F with respect to m, q, M and r, we obtain the saddle point equations
Fig. 2. The optimal loading rate ac as a function of l.
q ½1 bð1 qÞ2
We solve the equations numerically to obtain the phase diagram. We first investigate the noiseless limit T-0. Obviously, in this limit, q=1 holds. After some algebra, we find that the optimal loading rate ac is determined by the point at which the solution of the following equation with respect to y vanishes ( rffiffiffi 2 ) pffiffiffi 2 z y aþ ð1 lÞ exp p 2 ¼ ð1 lÞf1 2Hð yÞg where we defined the error function by Z 1 HðxÞ ¼ Dz x
In Fig. 2, we plot the optimal loading rate ac as a function of l. From this figure, we see that the optimal loading rate ac decreases monotonically. This means that the ferromagnetic retrieval order was destroyed by adding the anti-ferromagnetic term to the Hamiltonian. Thus, we conclude that if the components of the weight matrix of the networks are all positive, the plant intelligence model does not act as a memory device. Before we solve the saddle point equations for T 6¼ 0, it should be important to determine the
phase boundary between the spin glass and paramagnetic phases. The phase transition between these two phases is of first order, by expanding the saddle point equations around M=q=0, we obtain pffiffiffi the phase boundary line T SG ¼ ð1 lÞð1 þ aÞ. Now, we investigate the phase diagram for T 6¼ 0 by solving the saddle point equations numerically. We show the result in Fig. 3. From this figure, we find that the ferromagnetic retrieval phase shrinks to zero as the antiferromagnetic order increases as l-1. The behavior of the overlap M as a function of a is shown in Fig. 4. From this figure, we find that the overlap M becomes zero discontinuously at a=ac. We next consider the case of negative l. From the Hamiltonian, we find H¼
P 1X X l0 X m m x x S S SS i j i j ij ij i j N N m¼1
l0 ¼ lð40Þ As the parameter l increases, the system changes to the pure ferromagnet. Let us consider the limit of the parameter l ! 1 in the saddle point equation: R1 pffiffiffiffiffi M ¼ 1 Dz tanh b½ð1 lÞM þ z ar. Then, the term (1l)M appearing in the argument of tanh bð p Þ ffiffiffiffiffibecomes dominant, namely ð1 lÞ M z az even if the loading rateR a is large. 1 Consequently, equation M ¼ 1 Dz tanh
173 2.5 1.4 2
1
1.5
αc
1.2
T
0.8
T=0
1
0.6
0.5
0.4 0.2
λ=0.4
λ=0
λ=0.2
0 −3
0 0
0.02
0.04
0.06
0.08
0.1
0.12
−2.5
0.14
α
−2
−1.5
−1
−0.5
0
λ
Fig. 5. The optimal loading late ac as a function of l(o0) at T=0.
Fig. 3. The phase diagram of the system.
m
1 0.8
M
0.6 0.4 0.2
T=0.3
T=0.1
T=0.05
0 0
0.02
0.04
0.06
0.08
0.1
0.12
α Fig. 4. The overlap M as a function of a for the case of l = 0.2 at temperatures T = 0.05, 0.1, 0.3.
pffiffiffiffiffi b½ð1 lÞM þ z ar leads to M ¼ tanh b½ð1 lÞM Apparently, this equation has always a positive solution even if the temperature T=b1 is large. In this sense, the factor (1l) can be interpreted as temperature re-scaling. It is also possible for us to understand this result from a different point of view. In the saddle point equation Z 1 pffiffiffiffiffi M¼ Dz tanh b½ð1 lÞM þ z ar 1
the second pffiffiffiffiffi term appearing in the argument of tanh, z ar means cross-talk noise from the other
patterns ~ x ¼ ðxm1 ; . . . ; xmN Þ; m ¼ 2; ::; P, and obeys Gaussian distribution with zero mean and unit variance. On the other hand, the first term (1l)M 1 represents the signal of the p retrieval pattern ~ x. ffiffiffiffiffi Therefore, if the second term z ar is dominant, the 1 system cannot retrieve the embedded pattern ~ x. Usually, r in the second term grows rapidly as T increases. Obviously, if a increases, the noise term pffiffiffiffiffi z ar also increases. As a result, the signal part (1l)M becomes relatively small and the system moves from the retrieval phase to the spin glass phase. However, if l is negative and large, the signal part is dominant, and as a result, the noise part becomes vanishingly small. This is an intuitive reason why the optimal loading rate increases for negative l. In Fig. 5, we plot the optimal loading late ac as a function of lðo0Þ at T=0. As we mention, the optimal loading rate ac monotonically increases as l goes to N.
Concluding remarks In this chapter, we introduced a simple model based on a Hopfield-like network to explain the intelligence of plants. Inspired by the studies of Chakrabarti and Dutta (2003), we regarded a plant cell as a non-linear unit and investigated its corrective behavior as a network. In order to investigate the possibility of the plant network as a memory device, we constructed a mathematical
174
associative memory based on the Hopfield model in which ferromagnetic retrieval and antiferromagnetic terms co-exist. The strength of disturbance of pattern retrieval by the antiferromagnetic order is controlled by a single parameter. We found that the anti-ferromagnetic order prevents the system from recalling a pattern. This result means that the ability of the plant as a memory device is very weak if we set all weight connections to positive values. On the other hand, if we set all connections of the network to be negative, the capacity for memory increases. Our analysis in this paper was done for fully connected networks, and of course, for real plants, the cell membrane should be located in a finite dimensional lattice (Koyama, 2002) or in a scale-free network (Stauffer et al., 2003). So, extensive studies might be needed to investigate such ‘‘structure dependence’’ in the ability of the plant networks. Acknowledgments The author thanks Professor Bikas K. Chakrabarti for collaboration. This chapter was based on our publication (Inoue and Chakrabarti, 2005). He also acknowledges Professor Rahul Banerjee for good organizing the international workshop on Models of Brain and Mind: Physical, Computational and Psychological Approaches in Kolkata 2006.
References Ball, P. (2004). Do plants act like computers? Nature Science Update Weekly Highlights, 26 January 2004. Bose, I. and Karmakar, R. (2003) Simple models of plant learning and memory. Phys. Scr., T106: 9–12. Bose, J.C. (1923) The Nervous Mechanism of Plants. Longmans, London. Brueggemann, L.I., Pottosin, I.I. and Schoenkecht, G. (1998) Cytoplasmic polyamines block the fast-activating vacuolar cation channel. Plant J., 16(1): 101–105. Chakrabarti, B.K. and Dutta, O. (2003) An electrical network model of plant intelligence. Ind. J. Phys., A77: 551–556. Genoud, T. and Metraux, J.-P. (1999) Crosstalk in plant cell signaling: structure and function of the genetic network. Trends Plant Sci., 4(12): 503–507. Hertz, J., Krough, A. and Palmer, R.G. (1991) Introduction to the Theory of Neural Computation. Addison-Wesley Publishing, Cambridge, MA. Inoue, J. and Chakrabarti, B.K. (2005) Competition between ferro-retrival and anti-ferro orders in a Hopfield-like network model for plant intelligence. Phys. A, 346: 58–67. Koyama, S. (2002) Storage capacity of two-dimensional networks. Phys. Rev. E, 65: 016124-1–016124-6. Nishimori, H. (2001) Statistical Physics of Spin Glasses and Information Processing. Oxford University Press, New York. Peak, D.A., West, J.D., Messinger, S.M. and Mott, K.A. (2004) Evidence for complex, collective dynamics and emergent, distributed computation in plants. Proc. Acad. Sci. U.S.A., 101: 918–922. Stauffer, D., Aharony, A., da Fontoura Costa, L. and Adler, J. (2003) Efficient Hopfield pattern recognition on a scale-free neural network. Eur. Phys. J. B, 32: 395–399. Trewavas, A. (2002) Plant intelligence: mindless mastery. Nature, 415(6874): p. 841.
CHAPTER 14
A simple Hopfield-like cellular network model of plant intelligence Jun-ichi Inoue Complex Systems Engineering, Graduate School of Information Science and Technology, Hokkaido University, N14-W9, Kita-Ku, Sapporo 060-0814, Japan
Abstract: We introduce a simple Hopfield-like cellular-network model to explain a kind of ‘‘intelligence’’ in plants (Trewavas, 2002), especially, the capacity of plants to act as a memory device. Following earlier observations by Indian scientist J.C. Bose (1923), we regard the plant as a network in which each of the elements is connected via negative interactions. We investigate properties of the model by statistical mechanics. Keywords: plant intelligence; Hopfield model; associative memories; statistical-mechanical analysis; storage capacity; phase transition
the brain, is also based on parallel and distributed computation. Therefore, similarities between neural network models of brains and the plant networks should be discussed. Although the behavior of the dodder coil we mentioned above is due to the emergence of intelligence as a macroscopic function, it is important for us to investigate its rationale on a microscopic scale. Over 80 years ago, J.C. Bose (1923) detected electrical signaling between plant cells. Since then, following his experiments, many examples of cross-talk between biochemical signaling pathways in plants have been discovered. Especially, a Boolean representation of networks of signaling pathways is possible in terms of logical gates like AND, OR and XOR etc. These Boolean descriptions make it possible to draw analogies between plant networks and neural network models. Recently, Brueggemann et al. (1998) found the plant vacuolar membrane current–voltage characteristic to be equivalent to that of a Zenner diode. Inspired by their work, Chakrabarti and Dutta
Introduction Since pioneering studies by Indian scientist J.C. Bose (1923), plants have been regarded as a kind of network which are capable of intelligent responses to environmental stimuli. For example, the dodder coil, which is a plastic plant, explores a new host tree within hours subsequent to initial contact (Trewavas, 2002). This sort of behavior might be regarded as plant intelligence. If that is the case, does the plant compute, learn or memorize various spatial and temporal patterns in different environments similar to a computer or human brain (Ball, 2004)? Recently, Peak et al. (2004) pointed out that the plants may regulate their uptake and loss of gases by a distributed computation. As is well known, the ability of neural networks, which is a mathematical model of
Corresponding author. Tel.: +81-11-706-7225; Fax: +81-11-706-7391; E-mail: [email protected]
DOI: 10.1016/S0079-6123(07)68014-5
169
170
(2003) utilized the threshold behavior of plant cell membranes to develop or model gates for performing simple logical operations. They found that the plant network connections are all positive (excitatory by means of neuronal states) or all negative (inhibitory), compared to the randomly positive–negative distributed synaptic connections in real brains. As a result, the plant network does not involve any frustration in their computational capabilities and might lack the distributed parallel computational ability as in the case of associative memories. Even if we observe a single plant cell and provide a mathematical model for the single cell, we will not understand the macroscopic behavior of the plant giving rise to emergent properties like learning, memory and recognition (Genoud and Metraux, 1999; Bose and Karmakar, 2003). With this fact in mind, we investigate one of such collective behavior, namely, associative memories of plant networks, in this chapter. In other words, we try to answer the question ‘‘Do plant networks act as memory devices?’’ Especially, following the analysis by Chakrabarti and Dutta (2003), we investigate the equilibrium properties of the Hopfield model in which both ferromagnetic retrieval and anti-ferromagnetic terms co-exist. The strength of the anti-ferromagnetic order is controlled by a single parameter l. Within the replica symmetric calculation of statistical physics (Hertz et al., 1991; Nishimori, 2001), we obtain phase diagrams of the system. The l-dependence of the optimal loading rate ac, which is defined as the ratio of the maximum number of embedded patterns to the number of neurons in the brain, namely, Pmax/N, at T=0 is discussed. This chapter is organized as follows. In the next section, we introduce several experiments and observations related to the current–voltage characteristics of the plant cell membrane. In section ‘‘The Plant Intelligence Model’’, we model the plant possessing such properties by using a Hopfield-like network model in which both ferromagnetic-retrieval and anti-ferromagnetic terms exist. In section ‘‘Replica Symmetric Analysis’’, we analyze the model with the assistance of the replica method. In this section, we investigate to what extent the ferromagnetic retrieval order
remains against the anti-ferromagnetic disturbance. We also investigate the result of the ferromagnetic disturbance. All the results are summarized in the final section.
The I–V characteristics of cell membranes In this section, we briefly mention several results concerning properties of the plant units (cells), namely, current (I)–voltage (V) characteristics of their cell membrane. In Fig. 1, we show the typical non-linear I–V characteristics of cell membranes acting as logical gates. From this figure, we find that the I–V characteristics are equivalent to those of the Zenner diode. Namely, some threshold vT exists crossing which, the direction of the current changes. By assuming that the current has two directions, that is, 71, and the voltage is determined by the weighted contributions, Chakrabarti and Dutta (2003) constructed a mathematical plant cell as a non-linear unit. From the viewpoint of input– output logical units like perceptrons for neural networks, the output of the i-th unit Oi is given by ! N X wij I j Oi ¼ Y j¼1
where strength of each connection wij is all positive or negative, while in the Hopfield model it is given as 7 randomly distributed weight matrix in terms
1.5
0.5
5 mM
0 0
C2 vT(2) v
0
40 80 V (mV)
−0.5 −1
C1
1 mM
1
−60−40
l
l (A/m2)
vT(1) (a)
(b)
−1.5
Fig. 1. The non-linear I–V characteristics of cell membranes.
171
of the Hebbian rule wij ¼
P 1X xm xm N m¼1 i j
xmi 2 f1; 1g
From these experimental results and simple observations, we now ask a natural question, which is, could the plants act as memory devices similar to a real brain? Obviously, in the above definition of a single unit in the plant, there is no frustration as in animal brains. Thus, this paper attempts to elucidate, as to what extent constraints on the sign of the weight matrix influences the ability in plants to retrieve patterns as associative memories. For this purpose, we introduce a simple plant intelligence model based on a Hopfield-like model in which ferromagnetic retrieval and antiferromagnetic ordered phases co-exist. In the next section, we explain its details.
investigate the l-dependence of the system, namely, to study the l-dependence of the optimal loading rate ac(l) at T = 0 by using the technique of statistical mechanics for spin glasses.
Replica symmetric analysis In order to evaluate macroscopic properties of the system, we first evaluate the averaged free energy ½log Z~x ¼ ½log trS~ expðbHÞ~x where ½ ~x denotes the quenched average over the extensive P=aN patterns. To carry out this average and spin trace trS~ expðbHÞ, we use the replica method by using the relation ½log Z~x ¼ lim
n!0
The plant intelligence model We start from the following Hamiltonian ! P X 1X H¼ l xmi xmj Si S j ij N m¼1 ¼ H AF þ H FR where we defined the following two parts of the total Hamiltonian H AF ¼
lX Si Sj N ij
P 1 XX HF ¼ xm xm S i S j N ij m¼1 i j m In these expressions, the vector ~ x ¼ ðxm1 ; . . . ; xmN Þ is ~ ¼ ðS 1 ; :::; S N Þ stands m-th embedded pattern and S for neuronal states. A single parameter l determines the strength of the anti-ferromagnetic order, that is to say, in the limit of l-N, the system is completely determined by the energy function HAF. On the other hand, in the limit of l-0, the system becomes identical to the conventional Hopfield model. The purpose of this chapter is to
½Z n ~x 1 n
After standard algebra, we obtain the patternaveraged replicated partition function under the replica symmetric approximation as follows ½log Z~x ¼ extr expðnNFðm; q; M; rÞÞ m;q;M;r
with Fðm; q; M; rÞ ¼
b 2 bl 2 ab2 r M m þ ð1 qÞ 2 2 2 a bq þ log½1 bð1 qÞ 2 1 bð1 qÞ Z 1 pffiffiffiffiffi log Dz log 2 cosh bðlm þ arz þ MÞ 1
pffiffiffiffiffiffi where we define Dz ¼ dz expðz2 =2Þ= 2p. We should keep in mind that physical meanings of m and M are magnetization of the system, overlap ~ and a specific between the neuronal state S 1 recalling pattern ~ x among p=aN embedded patterns, respectively. The order parameter q stands for spin glass order parameters. In the next section, we evaluate the saddle point of this free energy density F and draw phase diagrams to specify the pattern retrieval properties of the system.
172 0.14
Phase diagrams
0.12
Z
pffiffiffiffiffi Dz tanh b½ð1 lÞM þ z ar ¼ m
1 1
0.06
0.02 0
pffiffiffiffiffi Dztanh b ð1 lÞM þ z ar
0
0.2
1
0.4
0.6
0.8
1
λ
2
q¼
r¼
0.08
0.04 1
M¼ Z
T=0 0.1 αc
In this section, we investigate the phase diagram of the system by solving the saddle point equations. By taking the derivatives of the free energy density F with respect to m, q, M and r, we obtain the saddle point equations
Fig. 2. The optimal loading rate ac as a function of l.
q ½1 bð1 qÞ2
We solve the equations numerically to obtain the phase diagram. We first investigate the noiseless limit T-0. Obviously, in this limit, q=1 holds. After some algebra, we find that the optimal loading rate ac is determined by the point at which the solution of the following equation with respect to y vanishes ( rffiffiffi 2 ) pffiffiffi 2 z y aþ ð1 lÞ exp p 2 ¼ ð1 lÞf1 2Hð yÞg where we defined the error function by Z 1 HðxÞ ¼ Dz x
In Fig. 2, we plot the optimal loading rate ac as a function of l. From this figure, we see that the optimal loading rate ac decreases monotonically. This means that the ferromagnetic retrieval order was destroyed by adding the anti-ferromagnetic term to the Hamiltonian. Thus, we conclude that if the components of the weight matrix of the networks are all positive, the plant intelligence model does not act as a memory device. Before we solve the saddle point equations for T 6¼ 0, it should be important to determine the
phase boundary between the spin glass and paramagnetic phases. The phase transition between these two phases is of first order, by expanding the saddle point equations around M=q=0, we obtain pffiffiffi the phase boundary line T SG ¼ ð1 lÞð1 þ aÞ. Now, we investigate the phase diagram for T 6¼ 0 by solving the saddle point equations numerically. We show the result in Fig. 3. From this figure, we find that the ferromagnetic retrieval phase shrinks to zero as the antiferromagnetic order increases as l-1. The behavior of the overlap M as a function of a is shown in Fig. 4. From this figure, we find that the overlap M becomes zero discontinuously at a=ac. We next consider the case of negative l. From the Hamiltonian, we find H¼
P 1X X l0 X m m x x S S SS i j i j ij ij i j N N m¼1
l0 ¼ lð40Þ As the parameter l increases, the system changes to the pure ferromagnet. Let us consider the limit of the parameter l ! 1 in the saddle point equation: R1 pffiffiffiffiffi M ¼ 1 Dz tanh b½ð1 lÞM þ z ar. Then, the term (1l)M appearing in the argument of tanh bð p Þ ffiffiffiffiffibecomes dominant, namely ð1 lÞ M z az even if the loading rateR a is large. 1 Consequently, equation M ¼ 1 Dz tanh
173 2.5 1.4 2
1
1.5
αc
1.2
T
0.8
T=0
1
0.6
0.5
0.4 0.2
λ=0.4
λ=0
λ=0.2
0 −3
0 0
0.02
0.04
0.06
0.08
0.1
0.12
−2.5
0.14
α
−2
−1.5
−1
−0.5
0
λ
Fig. 5. The optimal loading late ac as a function of l(o0) at T=0.
Fig. 3. The phase diagram of the system.
m
1 0.8
M
0.6 0.4 0.2
T=0.3
T=0.1
T=0.05
0 0
0.02
0.04
0.06
0.08
0.1
0.12
α Fig. 4. The overlap M as a function of a for the case of l = 0.2 at temperatures T = 0.05, 0.1, 0.3.
pffiffiffiffiffi b½ð1 lÞM þ z ar leads to M ¼ tanh b½ð1 lÞM Apparently, this equation has always a positive solution even if the temperature T=b1 is large. In this sense, the factor (1l) can be interpreted as temperature re-scaling. It is also possible for us to understand this result from a different point of view. In the saddle point equation Z 1 pffiffiffiffiffi M¼ Dz tanh b½ð1 lÞM þ z ar 1
the second pffiffiffiffiffi term appearing in the argument of tanh, z ar means cross-talk noise from the other
patterns ~ x ¼ ðxm1 ; . . . ; xmN Þ; m ¼ 2; ::; P, and obeys Gaussian distribution with zero mean and unit variance. On the other hand, the first term (1l)M 1 represents the signal of the p retrieval pattern ~ x. ffiffiffiffiffi Therefore, if the second term z ar is dominant, the 1 system cannot retrieve the embedded pattern ~ x. Usually, r in the second term grows rapidly as T increases. Obviously, if a increases, the noise term pffiffiffiffiffi z ar also increases. As a result, the signal part (1l)M becomes relatively small and the system moves from the retrieval phase to the spin glass phase. However, if l is negative and large, the signal part is dominant, and as a result, the noise part becomes vanishingly small. This is an intuitive reason why the optimal loading rate increases for negative l. In Fig. 5, we plot the optimal loading late ac as a function of lðo0Þ at T=0. As we mention, the optimal loading rate ac monotonically increases as l goes to N.
Concluding remarks In this chapter, we introduced a simple model based on a Hopfield-like network to explain the intelligence of plants. Inspired by the studies of Chakrabarti and Dutta (2003), we regarded a plant cell as a non-linear unit and investigated its corrective behavior as a network. In order to investigate the possibility of the plant network as a memory device, we constructed a mathematical
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associative memory based on the Hopfield model in which ferromagnetic retrieval and antiferromagnetic terms co-exist. The strength of disturbance of pattern retrieval by the antiferromagnetic order is controlled by a single parameter. We found that the anti-ferromagnetic order prevents the system from recalling a pattern. This result means that the ability of the plant as a memory device is very weak if we set all weight connections to positive values. On the other hand, if we set all connections of the network to be negative, the capacity for memory increases. Our analysis in this paper was done for fully connected networks, and of course, for real plants, the cell membrane should be located in a finite dimensional lattice (Koyama, 2002) or in a scale-free network (Stauffer et al., 2003). So, extensive studies might be needed to investigate such ‘‘structure dependence’’ in the ability of the plant networks. Acknowledgments The author thanks Professor Bikas K. Chakrabarti for collaboration. This chapter was based on our publication (Inoue and Chakrabarti, 2005). He also acknowledges Professor Rahul Banerjee for good organizing the international workshop on Models of Brain and Mind: Physical, Computational and Psychological Approaches in Kolkata 2006.
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