Text S11. The pre/post-spike IAF theory
To get a global account of the inverse gain sensitivity, we devised the pre/post-spike IAF
theory, which combines activation dynamics of the post-spike IAF theory with those of the
pre-spike IAF theory. Specifically, in the pre/post-spike IAF theory, the ISI is divided in two
consecutive periods. The membrane potential dynamics is governed first by the pre-spike IAF
theory (see equation 6.1, Text S6) in the time interval  0;T post  , then by the pre-spike IAF
theory (see equation 8.4, Text S8) in the time interval T post ;T  , where T  Tpost  Tpre is the
ISI duration. We denote Vpost the membrane potential at that time T post . We set Tpost  k post X ,
i.e. k post determines the relative number of activation time constants during which activation
exponentially relaxes toward x R (larger k post values result in a larger relative influence of the
post-spike IAF theory during the ISI). From equation (8.4), one can determine Vpost :



Vpost  VR  C 1   X gX xSP  xR VXR 1  e
 k post
 k
post
 X gL VLR  gX xR VXR  I  (11.1),
The membrane potential Vpost is taken as the initial condition of the pre-spike IAF theory, to
determine the time T pre required reaching VS , which is obtained from equation (6.2) by
replacing VR by Vpost :
1
T pre 
kX
1   0  x post I   S 
ln 

VS  Vpost I   0  xS I   post 


C 1 VS  Vpost

 I   
1
(11.2),
1
where x post denotes the steady-state activation at Vpost ,  0 , 1 and  S are those of the postspike IAF theory and  post  gL VLM  gX x post VXM . Thus, the firing frequency expresses as
1
f 
k post X 
kX
1   0  x post I   S 
1
ln 

VS  Vpost I   0  xS I   post 


C 1 VS  Vpost
1
 I   
1
1

(11.3).
The
firing
frequency
has


the
1
f 
form
u I 
k post X 
v I 
,
so
one
can
compute
2
k post X v  u
1
. After some algebra, one shows that the inverse gain can be
 

v u
 f
u  v
I
I I
1
written as


1
kX
1   0  x post I   S 
1
C VS  Vpost  1 
ln 
  k post X C VS  Vpost I  1 
 VS  Vpost I   0  xS I   post 


kX
1   0 
I  1   x post I   S   S   post I  1 
 1 

1
ln 

VS  Vpost I   0 
I   0   xS I   post  I   S  I   post 









2





(11.4).
For
gX  0 ,
the
term

1  0  gX VXM  0 ,
so

s 
 0  C VS  VR  1  k post X C 1 VS  Vpost  I  1  .
2
1

  0
cumbersome, so we computed s 
, i.e.
Again,
computing
that

gX
is
gX

2

 
 x post I   S 
1
k



1
X
1
0
 1 
ln
 k post X C VS  Vpost I  1  
  VS  Vpost I   0  xS I   post 
 


C VS  VR  








I




x
I






kX
1   0
I  1
s 
S
post
1
post
S




1
1
ln 

gX
VS  Vpost I   0 
I   0   xS I   post  I   S  I   post  





2
1


1
  1  k post X C VS  Vpost I  1 















(11.5)
to estimate its value. The input current was determined as in the pre-spike IAF theory. The
mean voltage potential was VM  1  pM VR  pM VS ,
p M being a fraction so that
VR  VM  VS . Best fits of the s landscape were obtained with input currents in the range
[0.25  1.5] nA1cm2 , and for kC values in the range 1;5 .
2