4.5 Asynchrony and WTA
4.5.3 Logical equations
The Thomas and D'Ari's method allows a precise stability analysis and predicts nicely nu- merical solutions. The main idea behind this method is to transform differential equations into
8 2 CHAPTER 4. Temporal Analysis logical equations. This transformation involves numerous steps which are now successively described for equations (4.27), (4.28), and (4.29).
* Given equation (4.27) and using the substitute (4.31) it comes:
* The linear term v in this latter equation must now be eliminated; this can be done by inserting the expression for v from equation (4.29):
dul - dv - 0:
* The steady solution of this latter equation is defined by - - - - dt dt
u, = w1f(u1)
-
w2 Cf(q) +f(u2)-
h2)f(v) - hl+
$1 ; (4.34)* Equation (4.34) is now ready to be transformed into a logical equation. For that purpose, definition of fix) is recalled: f(x) = 0 for x < 0 and fix) = 1 for x > 0
'.
Thus, x is directly expressed by a logical variable x which has the value 0 if x < 0 and the value 1 if x > 0. According to this principle, the differential equation (4.34) can be rewritten as follows:where U 1 is a logical function whose value reflects the evolution of the system, ul, u2, Y are logical variables, and duI(lexp) evaluate the location of the pseudo2 log- ical expression leXp in the scale of
threshold(^)^
of variable ul and is called "scaling".In this particular case (only one threshold), dul(le,) = 0 if I,,< 0 and dul(lexp) = 1 if lexp > 0.
This method of transformation is similarly applied on equations (4.28) and (4.29), resulting in the following logical equations:
Evaluation of the logical functions U1, U2, V requires consideration of every combination of the three corresponding variables ul, u2, v, leading to the so-called general stable table (Fig- ure 4.11). Logical expressions (KO, K1,
. . .
) are defined in Appendix 9. Note that two expres- sions can readily be evaluated (the underscore stands for state 0 or 1):V (0, 0, -) = dv(-h,) = 0, as h, is supposed to be always greater than zero. Using specific parameters for h l , h2, w l, w,, and setting conditions for the entries s l, s2, the general stable table can be reduced.
Ln
this example, the WTA structure has two entries. Parameters are chosen in order that 1. The case x = 0 can be treated, but in this case (x = 0 ) all one can say is that f(x) is in the range of values comprised in [0, 11.
2. Pseudo as real numbers can multiply boolean variables in these expressions.
3. In the general kinetic logic, a variable can have more than one threshold.
Asynchrony and WTA 8 3 only one signal wins the competition. Note that the logical state of the winner is 1, and the log- ical state of the loser is 0 (ul or u2). Parameters are thus chosen to obtain in equation (3.66) the condition K = 1. For instance, if parameters h,, h2 are set to value 0.5 and parameters w l , w2 are set to value 1, it comes:
With this condition and given that the amplitude of the loser will be restricted to remain less than 1, there will be only one winner (K < 2).
Figure 4.1 1 : General stable table. This table is the evaluation of the logical equations (4.33, (4.36), and (4.37) for all possible combinations of logical variables ul, u 2 , and v
.
Expressions KO to KI3 are defined in Appendix 9.To compare the effects on the dynamics of the WTA neuronal structure obtained from syn- chronous inputs with those obtained from asynchronous inputs, two cases are considered: (i) identical inputs (sl = s2); (ii) differing inputs (e.g. s, > s 2 ) In the first case, signals should occur at the same time, whereas in the second case, signal s, should have the temporal prece- dence on signal s2 (according to section 4.1). These two cases are successively treated in sec- tions 4.5.5 and 4.5.7 using both, an asynchronous and synchronous analysis.
4.5.4 VVTA stable table for identical input signal amplitudes
The general stable table, shown in Figure 4.11, is now evaluated for the particular case s, = s2 = s < 1 and s > h l , h 2 (where h = h2 = 0.5). Results of this evaluation are shown in Figure 4.12. This table represents the first step in the temporal analysis of the neuronal struc- ture. The second step will be the graph of transitions. Before considering this graph, some ex- planations must be given on the stable table.
First, an overline above a number indicates that the logical variable value differs from the logical function value (for an example, see legend in Figure 4.12). As a consequence of this disparity, an imminent transition is expected to happen. In the particular situation where a line of the table contains more than one imminent transition, they are treated successively in case the asynchronous analysis is applied. Simultaneous transitions are not excluded but probability
84 CHAPTER 4. Temporal Analysis for their occurrence is supposedly negligeable. Thus, the various temporal possibilities must be explored.
Second, an ellipse drawn around a line formed by logical variable values, indicates that no transition is going to happen. Such a state is called a stable node (whence the name "stable ta- ble"). Stable nodes correspond to stable solutions of the differential equations.
Figure 4.12 : Stable table corresponding to the evaluation of expressions KO to K13, defined in Appendix 9, for the following parameters: hl = h2 = 0.5, w 1 = w 2 = 1, and sl = s2 = 0.8. Every time the value of a logical variable is differing with the logical function value, an overline is indicated on the former. For instance, in the frst line, the value of the logical variables are 000 whereas the value of the logical functions are 110. Thus, this situ- ation is indicated by WO. When every logical variable value is identical to its corresponding logical function value, an ellipse is drawn around the logical values.
4.5.5 WTA graph of transitions for identical input signal amplitudes
A convenient representation for analyzing the stable tables is the graph of transitions1. In such a graph (see for instance Figure 4.13), a vertex represents the state of the logical variables and a link represents a transition (only one transition for the asynchronous case, one and more transitions for the synchronous case). Stable nodes are surrounded by an ellipse.The method to build a graph of transitions from a stable table can be summarized as follows: (i) take the line of the table corresponding to the initial conditions (generally, the first line); this is the fust ver- tex; (ii) when only asynchronous transitions are authorized, consider the logical value with an overline and take its complementary value (0 to 1, 1 to 0); (iii) read in the table the correspond- ing logical variable states and up-to-date the overlines; this is the second vertex; (iv) draw a link between the initial and second vertex; (v) repeat the method iteratively until a stable node is reached.
The graph of transitions is now drawn for two cases: (i) asynchronous; (ii) synchronous. In the first case, no simultaneous transitions are permitted. Alternatively, in the second case, only
1. Such an analysis is to be found in Thomas and D' Ari (1990) but also in classical books dealing with logic (e.g. chapter 7, Mange 1979).
Asynchrony and WTA 8 5 synchronous transitions are allowed. The initial state is supposed to be 000, corresponding to the logical variable u l = u2 = v = 0. Note that the results of this logical analysis will be compared to the analogical solution, performed by numerical evaluation using the Runge-Kutta method.
Asynchronous graph of transitions
Remember that both input signals have the same amplitude and, in consequence, should have the same latency. Nevertheless, an asynchronous analysis is performed to illustrate the ef- fect of asynchronous transitions. In this context, the graph of transitions shown in Figure 4.13 is obtained.
One determinant parameter in this graph of transitions is clearly whether signal s l occurs before or after signal s2. The former situation corresponds to the case where latency of signal s, is shorter than latency of signal s2 (11 < Z2). The second situation corresponds to the oppo- site case (12 c ll). Also, as both entries have their amplitude greater than threshold hl and h2, whenever they occur, the elementary units become active (ul or u2). Apart form these tempo- ral conditions on the entries, the graph of transitions shows other conditions which are "inter- nal" to the neuronal structure and depend on the time constant of the elementary units. Accord- ing to this graph, there are two possible stable nodes: u l = 1, u2 = 0, and u2 = 1, ul = 0, for v = 1.
Figure 4.13 : Asynchronous graph of transitions corresponding to the stable table shown in Figure 4.12. From the initial state, there are two possibilities whether signal sl occurs before signal s2 and vice versa (the former pos- sibility is indicated by relation 1 I < i2). Note that these two possibilities are artificial as signals have, supposedly, the same latencies. There are two possible stable nodes: 01 1 and 101.
Synchronous graph of transitions
By considering signals occurring both at the same time (which is the mathematical condi- tion if their amplitudes are identical), and only simultaneous transitions as possible (synchro- nous analysis), a reduced graph of transitions is obtained (Figure 4.14). Interestingly, the re- sulting graph contains a cycle. Thus, solutions will change constantly over time. Although syn- chrony is only artificial and in most cases biological, system keep slight time differences sufficient to avoid such periodic solutions, asynchrony in neuronal structures can also lead to oscillatory modes (Durante and Burgi 1992; see also section 3.5).
CHAPTER 4. Temporal Analysis
Figure 4.14 : Synchronous graph of transitions corresponding to the stable table shown in Figure 4.12. From the initial state, there is only one possibility when the transitions occur simultaneously (because both signals occurs simultaneously). In this case, a cycle results. Note that it may have 1, 2, or 3 simultaneous transitions. The dashed line is referring to a cycle produced when only the logical variables ul and u2 are allowed to change synchro- nously, a condition found in the numerical solution shown in Figure 4.16.
4.5.6 WTA simulations for identical input signal amplitudes
An important characteristic of this logical analysis presented in this section is the good prediction of the numerical solutions it allows. Apart from the information on steady states (sta- ble nodes, cycles), it is possible to obtain a numerical evaluation of the solution. For instance, the state 10 1 gives the approximative values u l = 0.8, u2 = -0.2, and v = 0.5 by using equations (A64), (A65), and (A57), respectively, without the scaling operator (which discretiz- es analogical values). In the same way, the state 011 gives the approximative values u , = -0.2, u , = 0.8, and v = 0.5 (equations (A62), (A63), and (A57), respectively). Results of the numerical solutions can be are found in Figure 4.15 and Figure 4.15.
The numerical values for asynchronous entries are indeed in agreement with the predic- tions. Note that only the result corresponding to the case lI < l2 is shown in Figure 4.15, as con- dition l2 < lI leads to the same solutions if indices 1 and 2 of variables u i are inverted.
Figure 4.15 : Simulations of the WTA structure shown in Figure 3.25, for two entries, with precedence of sl on s2. Parameters are: time interval dt = 0.5 msec ; time constant .c = 1000 ms ; thresholds hl = h2 = 0.5 ; syn- aptic weights wl = w2 = 1 ; entries sl = 0.8 at t 2 @ and s2 = 0.8 at t 2. 100 ms . Steady state is ul = 0.8,
u2 = -0.2, v = 0.5. The unit for thresholds, entries and steady states is Volt. Precedence of entry s2 on sl is not drawn but results in similar curves, whose asymptotic solution is the stable node 01 1 instead of 101.
The numerical solution for synchronous entries, shown in Figure 4.16, demonstrates a pe- riodic solution corresponding to the cycle in the graph of transitions. In this simulation, transi- tion 001 to 110 does not occur. Synchronous transitions appear only with the simultaneous
Asynchrony and WTA 87 transitions of the state of the elementary units referred to as u l and u2 (initiated by synchro- nous entries sl and s 2 ) Then, time constant T introduces differing delays for the elementary unit referred to as v. In consequence, and according to the time constant z, two cycles are pro- duced: 000, 1 10, 1 1 1 , 00 1 , 000, . . . and 1 1 1, 00 1, 1 11, .
. . .
These two cycles are represented by dashed Lines in the graph of transitions in Figure 4.14time [s]
- 1
Figure 4.16 : Simulations of the WTA structure, for two synchronous entries. Parameters as in Figure 4.15, except for the entries: sl = 0.8 and s2 = 0.8 at t 2 0 . The resulting cycle is equivalent to the states 000,110,111,001, 000, ...
4.5.7 WTA stable table for differing input signal amplitudes
The general stable state is now evaluated for the particular case s, > s2, s1 > 1 , and 0.5 < s2 < 1 . Results of this evaluation are to be found in Figure 4.17. This table indicates that only one stable node exists (only one line enclosed in an ellispe), As before, the asynchronous and synchronous cases are analyzed by using the graphs of transitions.
Figure 4.17 : Stable table correspon&ng to the evaluation of expressions KO to K13, defined in Appendix 9, for the following parameters: hl = h2 = 0.5, wl = w2 = 1 , and sl = 1.2, s2 = 0.8. Explanations for the over- lines and ellipse are given in Figure 4.12.
88 CHAPTER 4. Temporal Analysis