Table S1. Network parameters
Parameter
τ
α
β1
β2
γ
τad
T1,2
T3
Single WTA
Value
0.2
1.2
3
0.2
0.5 (model 2)
0 (model 1)
200
6
0.5
Parameter
τ
α
β1
β2
φ
Double WTA
Value
0.2
1.4
3
0.2
0.25
t
tI
[0.5 0.5 0.5]
[6 6 0.5]
Online Methods
Single WTA (model 1 and 2)
The dynamics of the single WTA models (model 1 and model 2) are implemented using the
following differential equations:
du
 j  Ru j  f ( I j   u j  1u3  T j   A)
dt
dAj
 ad
  Aj  u j
for j = 1,2
dt
2
du
 3  Ru3  f (  2  um  T3 )
dt
m 1
Where τ is the integration time constant, R is the load or conductance and is set R = 1. Ij(t) is
the external input to unit j, α, β1 and β2 are positive (constant) parameters. γ is the selfadaptation gain and τad is the adaptation time constant which is much slower than τ. The
thresholds Tj are positive, equal for all units and constant in this circuit. f(x) is a nonsaturating rectification non-linearity f(x) = max(0,x). In the single WTA without adaptation
(model 1), γ is set to zero. All other values are identical in model 1 and 2 and given in table 1.
Inputs applied to u1 and u2 have a Gaussian distribution, whose mean (“input strength”) is
fixed in each simulation and whose standard deviation is set to 1 independent of input
strength. For models 1 and 2, input strengths in a range of 6.0 to 7.0 are tested in steps of 0.1.
Double WTA (model 3)
For model 3, the two coupled WTA circuits I and P are described by the following coupled
differential equations:
P 
I 
dP
 f (W  P  X 1  I  t )
dt
dI
 f (W  I  X 2  P  Inp  tI )
dt
Upper case letters indicate matrices and lower case letters with arrows indicate vectors. The
matrices describing the connection strengths within and between the circuits are:
  0   1
 0 0
 0  0




W   0    1 , X 1   0  0  , X 2   0 0
  2  2 0 
 0 0 0
 0 0 0
Here, τ is the same time constant as in the single WTA, W is the weight matrix of all single
1
WTA circuits (note: the parameters differ between single and double WTA). X1 contains
feedforward weights from circuit I to circuit P while X2 contains the weights of feedforward
input from P to I. Parameters used for all simulations are given in table 1.
External inputs are applied to i1 and i2 and follow a Gaussian distribution as in the case of
models 1 and 2. Input strengths reach from 5.0 to 6.0 in steps of 0.1.
Simulations
Numerical integration was done using Euler integration with an integration time constant of
δ=0.001. After every integration step, the nonlinear rectification function f(x) = max(x,0) was
applied to each circuit.
The parameters were chosen such that stability is guaranteed. The criteria for stability,
obtained by contraction analysis, are43:
0    2  1 2
0 2
0   1 2  1
  2  1 2  
Changes in these parameter values can modulate median dominance duration and the spread
of the distribution. Since the focus of the present paper is on the effect of varying input
strength, rather than on the parameters governing network dynamics, we chose a fixed
parameter set for each condition (single and double WTA) such that the systems remain stable
and show alternating behavior for a range of input strengths. The chosen parameter values
1
(table 1) are in the range typically used 43,48 and result in a high gain G ( G 
).
1   1 2  
Noise is updated independently for each input unit every 100 integration steps. The exact
noise update frequency is not critical, given that it is substantially larger than the integration
time constant δ and substantially smaller than τ 48. As long as these criteria are fulfilled, the
value of the time constant τ is also uncritical and was here chosen to be 0.2 for all simulations
for reasons of numerical efficiency. A natural time unit for all simulations is obtained by
factoring out the arbitrary δ from the number of integration steps and is thereby given in units
of 1/ δ (i.e., 1000 integration steps). All times reported for simulations are given in this unit.
Each simulation was run for 1,500,000 integration steps; that is, for 1500 units of time and
repeated five times with different random values for the noise.
2