Bioinformatics An Introduction 4th Edition (Jeremy Ramsden)

12.3 Synergetics

153

Fig. 12.3 Phase portrait of the system represented by Eq. (12.29) with a equals 1 divided by 3a = 1/3. The main

isoclines (cf. Eq. 12.29) are u 1 equals 0u1 = 0 and u 2 equals 1 minus a u 1u2 = 1 −au1 (“vertical”, determined from upper F 1 equals u 1 minus u 1 u 2 minus a u 1 squared equals 0F1 = u1 −

u1u2 −au²

1 ⁼^{0 with}^{Delta u 1 equals 0}^Δ^u¹⁼^{0), and}^{u 2 equals 0}^u²⁼^{0 and}^{u 1 equals 1 minus a u 2}^u¹⁼¹⁻^au²^{(“horizontal”, determined from}

upper F 2 equals u 2 minus u 1 u 2 minus a u 2 squared equals 0F2 = u2 −u1u2 −au²

2 ⁼^{0 with}^{Delta u 2 equals 0}^Δ^u²⁼^{0), shown by dashed lines. The system has four stationary}

states: atu 1 equals u 2 equals 0u1 = u2 = 0, unstable,lamda 1 equals lamda 2 equals plus 1λ1 = λ2 = +1; atu 1 equals u 2 equals 1 divided by left parenthesis 1 plus a right parenthesisu1 = u2 = 1/(1 + a), unstable (saddle point),

lamda 1 equals negative 1 comma lamda 2 equals left parenthesis 1 minus a right parenthesis divided by left parenthesis 1 plus a right parenthesis greater than 0λ1 = −1, λ2 = (1 −a)/(1 + a) > 0;atu 1 equals 1 divided by a comma u 2 equals 0u1 = 1/a, u2 = 0,stable,lamda less than 0λ < 0;and atu 2 equals 1 divided by a comma u 1 equals 0u2 = 1/a, u1 = 0,

stable, lamda less than 0λ < 0. The separatrix (separating the basins of attraction) is shown by the dashed-dotted

line (after Chernavsky 1990)

Habituation

Empirical observation of many systems over time reveals that their responses to

regularly repeated stimuli over time tend to decrease. This is called habituation or,

especially when observed in a living system, fatigue. At the ﬁrst sight this might

seem paradoxical: it may be supposed that most real systems, of which the example

in Sect. 12.3 is a simple illustration, are multistable and, hence, should potentially

display considerable variety of behaviour. The explanation is that no matter how rich

a system may be in states of equilibrium, after a time it will typically be found to

be in a single basin of attraction. ²²Although both an initial endowment of potential

variety of behaviour and ultimate stability seem like very necessary attributes for a

cell whose fate is to be a highly differentiated member of an organ, in other cases

22 See Ashby (1958) for a proof.