Point Neurons with Conductance-Based Synapses in
the Neural Engineering Framework

arXiv:1710.07659v1 [q-bio.NC] 20 Oct 2017

Andreas Stöckel, Aaron R. Voelker, and Chris Eliasmith
Centre for Theoretical Neuroscience, University of Waterloo
October 24, 2017

Abstract
The mathematical model underlying the Neural Engineering Framework (NEF)
expresses neuronal input as a linear combination of synaptic currents. However, in
biology, synapses are not perfect current sources and are thus nonlinear. Detailed
synapse models are based on channel conductances instead of currents, which require independent handling of excitatory and inhibitory synapses. This, in particular,
significantly affects the influence of inhibitory signals on the neuronal dynamics.
In this technical report we first summarize the relevant portions of the NEF and
conductance-based synapse models. We then discuss a naïve translation between
populations of LIF neurons with current- and conductance-based synapses based
on an estimation of an average membrane potential. Experiments show that this
simple approach works relatively well for feed-forward communication channels,
yet performance degrades for NEF networks describing more complex dynamics,
such as integration.

1 Introduction
As a central assumption, the Neural Engineering Framework (NEF, [1]) relies on linear
interaction of pre-synaptic activity in the form of synaptic currents injected into the
neuron membrane. This assumption is readily fulfilled by current-based synapse models.
Biological synapses however are not perfect current sources. Instead, currents injected
into the cell membrane are caused by changes in receptor channel conductance directly or
indirectly gated by neurotransmitters released in the pre-synapse. Consequently, current
flow into the membrane is modulated both by pre-synaptic spikes, as well as—due to the
nature of ion channels—the difference between the neuron membrane potential and the

1

specific ion species reversal potential. This nonlinear interdependence is modelled by
conductance-based synapse models [2, 3].1
For various reasons, conductance-based synapses are a useful addition to the NEF.
Foremost, when modelling functional, biologically plausible neural networks, more realistic synapse models pose additional constraints on the network design and reduce degrees
of freedom. For example, conductance-based synapses strictly differentiate between excitatory and inhibitory synapses, allowing the network designer to map their parameters
to those derived from neurophysiological evidence. Furthermore, some neuromorphic
hardware systems (e.g. Spikey, BrainScaleS; [5, 6]) exclusively implement conductancebased synapses, which must be taken into account when assembling networks for these
platforms. Finally, nonlinear synaptic interaction has been argued to be an important
element of biological neural networks, and as a site for biological computation [2]. Exploiting these nonlinearities in the NEF increases the computational power per neuron.
In the future, conductance-based synapses may thus be utilized to increase the efficiency
of certain mathematical operations such as multiplication, requiring fewer neurons at the
same precision.
The remainder of this report is structured as follows: section 2 gives a short overview
of the relevant equations in the NEF. Section 3 summarizes the biological background of
conductance-based synapses and describes the underlying theoretical model. Section 4
elaborates on the actual integration of conductance-based synapses into the NEF and
provides an equation highly similar to the original NEF formalism, yet containing additional, voltage-dependent scaling factors. Section 5 provides estimations for these factors
based on the expected value for the average membrane potential. Section 7 presents two
benchmark experiments, followed by concluding remarks in section 8.

2 Current sources in the Neural Engineering Framework
The NEF is a systematic approach to the representation of low-dimensional vectors in
populations of spiking neurons and the approximation of arbitrary functions in the connections between these populations [1]. As an example, consider two populations of n
spiking neurons with all-to-all connectivity from the first (pre-synaptic) to the second
(post-synaptic) population. Let ~
x (t ) be the value represented by the pre-synaptic population, and ~
y (t ) the value represented by the post-synaptic population. Given a function f ,
the NEF allows us to compute connection weights w i j such that the resulting activity in
the post-population represents ~
y (t ) ≈ f (~
x (t )).
1 Not to be confused with conductance-based neuron models, such as the Hodgkin-Huxley model of action
potential generation [4, 2].

2

The formalism underlying the NEF assumes that each neuron j receives its input
in the form of a current J (t ) injected into its (virtual) neuron membrane.2 This current
is the sum of a constant bias current J bias , and a synaptic current J syn (t ). The latter is
calculated from a weighted sum of filtered input spike events: spikes produced by the i th
pre-synaptic neuron are modelled as a sum of Dirac pulses a i (t ), and a filter h(t 0 ) defines
the shape of the post-synaptic current in response to a single input spike:
J (t ) = J bias + J syn (t ) = J bias +

n
X
i =1

w i j (a i ∗ h)(t ) .

(1)

Central to the NEF is the idea of factorizing the connection weight matrix W into
e j , where w i j =
gain factors α j , decoding vectors d~i , and unit-length encoding vectors ~

α j 〈~
e j , d~i 〉. Correspondingly, eq. (1) can be written as
J (t ) = J bias + J syn (t ) = J bias + α j ~
e Tj

³X
n
i =1

´
d~i (a i ∗ h)(t ) .

(2)

The decoders d~i linearly project the high-dimensional activity of the pre-synaptic population onto the desired low-dimensional representation ~
y (t ) = f (~
x (t )). Multiplication

with unit-length encoders ~
e j corresponds to a projection of this low-dimensional representation onto the preferred direction of each post-synaptic neuron j . Correspondingly,
the transformed value is again represented as a high-dimensional population coding,
allowing to chain an arbitrary number of neuron populations and transformations.
As a side-effect, factorization of the weight matrix W into decoders and encoders
simplifies the calculation of w i j . Instead of optimizing weights w i j individually, we
can randomly select bias currents J bias
, gain factors α j , and encoders ~
e j , and calculate
j
decoders d~j from a least squares linear regression
¡
¢−1
Y = D A ⇒ D ≈ Y A T A A T + I σ2
,

(3)

where the matrix D is composed of the decoding vectors, A is a matrix of neuron responses
in the post-synaptic population for a set of sample points, and Y is a matrix of the desired
function outputs for these points. The expression I σ2 is a regularization term which
accounts for uncertainty and noise in the neuron response.
Crucially, the above equations contain the implicit assumption that the input current
J (t )—and in turn the effect of each individual input spike—is independent of the neuron
state at time t . Furthermore, all input spikes are treated equally (except for the scaling
factor w i j ), and there is no distinction between excitatory and inhibitory synapses. Having
2 Note that J (t ) (as well as other currents and conductances) should be read with a subscript j in mind – all

equations, unless explicitly stated otherwise, refer to the j th neuron in the post-synaptic population.

3

gL
J syn (t )

J bias

Cm

v(t )

EL

Figure 1: Equivalent circuit diagram of a LIF neuron with current-based synapses (subthreshold
behaviour; without spike generation/reset mechanism).

synaptic currents which are independent of the neuron state, is commonly referred to as
using “current-based synapses” [2, 7].
The subthreshold circuit diagram of a LIF neuron with current-based synapses is
depicted in fig. 1. The dynamical system describing this circuit is3
Cm

dv(t )
= g L (E L − v(t )) + J bias + J syn (t ) ,
dt

(4)

which has a stable attractor (the equilibrium potential E eq ) at
E eq (t ) =

J bias + J syn (t )
+ EL .
gL

(5)

The synaptic current J syn (t ) drives the membrane potential v(t ) towards more positive
values for input spikes received at excitatory synapses (w i j > 0) and towards more negative

values for spikes received at inhibitory synapses (w i j < 0). Importantly, the equilibrium

potential E eq in eq. (5) can be driven to arbitrary values if the input currents J are large
enough. In practice this is not an issue for positive J , as the maximum membrane potential
is limited by the neuron threshold potential v th . However, such a natural limitation does
not exist for negative currents, and the NEF puts no constraint on the negative current
strength, for neither J bias , nor J syn . A solution to this problem is to modify the LIF model,
such that the membrane potential is clamped to a minimum value4 , i.e., the reset potential
v reset . Note that all experiments in this report are based on LIF neurons with such a
minimum membrane potential.
This problem does not occur for conductance-based synapses. Conductance-based
synapses are not only biologically motivated, but eliminate the physically implausible,
3 The ratio between the membrane capacitance and the leak channel conductance is the membrane time

constant τRC = C m /g L .
4 This is for example implemented in the LIF neuron model in the software neuron simulator included in
Nengo, the reference software implementation of the NEF.

4

(-)

v

(+)

v

(-)

+ +
+ +
- - - - +

+ + + +
- - - -

220 mosm

(a) Membrane without K+ -channel

-

10 mmol K+
100 mmol Na+
110 mmol Cl-

10 mmol K+
100 mmol Na+
110 mmol Cl-

+

K+
Na+
ClA-

+ + +
+
- - -

100 mmol
10 mmol
10 mmol
100 mmol

Osmotic/electrical
forces

+

+ + +
+
+ - - - - 100 mmol K+
10 mmol Na+
10 mmol Cl100 mmol A+

(+)

K+ channel

(b) Added K+ -channel

Figure 2: Illustration of the ion channel reversal potential caused by equalling osmotic and electric
forces. Although the intra- and extracellular fluids are electrically neutral (a), an open, selectively
permeable ion channel causes a change in the membrane potential (b). Illustration adapted from
[8]. See [2, 9] for more details.

perfect current sources altogether. Unfortunately, and what will be the focus of the
remainder of this report, conductance-based synapses introduce a dependency between
the neuron membrane potential v(t ), and the injected current J syn (t ). As a result, the post
synaptic currents (PSCs) of pre-synaptic spikes no longer combine linearly. Consider two
excitatory input spikes which do not trigger an action potential at the receiving neuron.
Due to the increased membrane potential v(t ), the PSC induced by the second spike
is smaller than the PSC induced by the first spike. Furthermore, conductance-based
synapses require excitatory and inhibitory synapses to be handled differently. Both issues
are not accounted for in the standard NEF eqs. (1) and (2).

3 Conductance-based synapses
In biology, changes in the neuron membrane potential are primarily caused by variations in the permeability—or conductance—of selective ion channels embedded into the
lipid bilayer cell membrane [2, 9]. As depicted in fig. 2a, ions are dissolved in different
concentrations in the intra- and extracellular fluid. Ion channels provide a way for a
specific ion species to follow the osmotic concentration gradient, which in turn causes
a charge imbalance and an electrical force countering the osmotic force (fig. 2b). The
charge imbalance itself is measurable as a change in the membrane potential v(t ). Once a
certain membrane potential is reached, the osmotic and electric forces cancel and the ion
flow stops. Further increase/decrease of v(t ) reverses the ion flow through a particular
5

gL

g Na+ (t )

g K+ (t )
Cm

EEL Na+

v(t )

E K+

Figure 3: Equivalent circuit diagram of a LIF neuron with conductance-based synapses (without
spike generation/reset mechanism), see also [2].

channel, giving rise to the name reversal potential for this particular v(t ).5 The exact value
of the reversal potential differs between ion channel types. For example, in mammalian
+
cells, the reversal potential for a sodium channel is E Na
≈ 67 mV, while for potassium the

reversal potential is E K+ ≈ −83 mV. In the resting state, the membrane is slightly permeable
for a mixture of ions, resulting in an equilibrium potential6 of E L ≈ −65 mV [3, 10].

An equivalent circuit diagram of a cell membrane popularized by Hodgkin and Hux-

ley [4] (though in the context of action potential generation, and not synaptic input)
with leak, sodium and potassium channel is shown in fig. 3. Post-synaptic currents can
be modeled as a change of permeabilies in the Na+ and K+ ion channel.7 For example,
excitatory input spikes could cause the sodium channels to be open for a certain period of
time, driving the membrane potential towards E Na+ . Conversely, inhibitory input spikes
might cause potassium channels to open, driving the membrane potential towards E K+ .
In the following, we refer to E Na+ as E E (the excitatory reversal potential) and to E K+ as
E I (the inhibitory reversal potential). The dynamics for conductance-based synapses in
response to incoming spikes a(t ) can be described with a linear filter h(t 0 ), where the
channel conductance g (t ) is g (t ) = (a ∗ h)(t ). Possible filters include simple exponential

filters, the alpha function, or the difference of exponentials [7]. Of course, in biology,
channel dynamics involve the release of a neurotransmitter, diffusion of the transmitter through the synaptic cleft, and a chemical cascade causing ion channels to open as
soon as the transmitter reaches the post-synaptic neuron. More detailed models of these
highly nonlinear processes exist [7]. Nevertheless, in the remainder of this report we
5 Since ion diffusion is a stochastic process, ions will always diffuse in and out of the cell as long as the
channel is open. The above claims holds for the average ion flow only.
6 To clarify the difference between reversal and equilibrium potential: the reversal potential is the potential
at which the flow of a single ion species reverses direction, the equilibrium potential is the potential at which
the sum of all currents is zero.
7 In biology, a large number of channels are embedded into the neuron cell membrane. Individual ion
channels are either fully open or fully closed, following a stochastic process. The conductance g (t ) can be
interpreted as the probability of individual channels to be open.

6

assume the synaptic dynamics to be describable as a linear filter, and in particular focus
on normalized exponential low-pass filters with time constant τsyn where

h(t ) =




1
τsyn

exp(−t /τsyn ) t ≥ 0

0

t <0

,

and

´
n
X
dg (t )
1 ³
w i j a i (t ) .
=−
g (t ) +
dt
τsyn
i =1

Note that while the synaptic dynamics are linear with respect to the input, the post
synaptic current is not (see section 2).
The differential equation describing a LIF point neuron with conductance-based
channels is
Cm

dv(t )
= g L · (E L − v(t )) + g E (t ) · (E E − v(t )) + g I (t ) · (E I − v(t )) .
dt

(6)

The equilibrium potential E eq (t ) for this model is a weighted average of the individual
reversal potentials
E eq (t ) =

g L · E L + g E (t ) · E E + g I (t ) · E I
.
g L + g E (t ) + g I (t )

(7)

This implies that—in contrast to current-based models, which, as noted at the end of the
previous section, can provide arbitrarily large input—the membrane potential can never
be driven to values outside the voltage range spanned by E L , E E , and E I , or, in biological
terms, the ion channel reversal potentials.

4 Conductance-based synapses in the NEF
One of the initial challenges when implementing conductance-based synapses in the NEF
is the different handling of excitatory and inhibitory synapses. Spikes arriving at excitatory
synapses will influence the excitatory ion channel conductance, whereas spikes arriving
at inhibitory synapses influence the inhibitory ion channel conductance.
This issue is virtually nonexistent when adhering to Dale’s Principle [11]. Each prepopulation of neurons will either project with excitatory or inhibitory connections to the
post-population, and influence one of the two channel types only.
In the general case, the weight matrix W (section 2) can be split into two non-negative
matrices W + and W − . The matrix W + contains positive, excitatory weights, whereas W −
contains negative, inhibitory weights with all other weights set to zero:
w i+j = max{0, w i j } ,

w i−j = max{0, −w i j } .

7

This non-negative decomposition of W unfortunately ruins the factorization into
encoders and decoders employed by the NEF. For two reasons, this is not a real issue.
First, decomposition happens after encoders and decoders have already been computed,
maintaining the entire NEF pipeline. Second, to increase the computational efficiency
during simulation, artificial decoder and encoder matrices E ± , D ± can be calculated,
e.g., by performing a singular value decomposition of W + and W − :
~ ± = U [k] ,
E

W ± = U SV ,

~ ± = V [k] · S[k] .
D

Here, k is the number of non-zero eigenvalues or, alternatively, eigenvalues above a
certain threshold, and M [k] refers to the first k columns in the matrix M . Under most
circumstances, this factorization yields little to no error, since the weight matrix W is
an outer product and thus of low rank. Usually, k is equal to the number of dimensions
represented by the population plus one (caused by clamping the weights).
As mentioned in the previous section, the channel conductance time-dynamics g E (t )
and g I (t ) can be modeled as a linear filter of the input spike trains. Essentially, we can
just reuse the post-synaptic current filter h(t ) from eq. (1) and apply it to the individual
channel conductances instead. Assuming (as in section 2 and without loss of generality)
that all synapses have the same synaptic filter h(t ), the conductances g E (t ) and g I (t ) for
the j th neuron are
g E (t ) =

n
X
i =1

w i+j · (a i ∗ h)(t ) ,

g I (t ) =

n
X
i =1

w i−j · (a i ∗ h)(t ) .

(8)

As per eq. (6) the synaptic current J syn (t ) injected into a single neuron j with conductancebased synapses is
J syn (t ) = g E (t ) · (E E − v(t )) + g I (t ) · (E I − v(t )) .

(9)

Combing eqs. (8) and (9) yields
J syn (t ) =

n ³
X
i =1

´
w i+j · (E E − v(t )) + w i−j · (E I − v(t )) · (a i ∗ h)(t ) .

(10)

This equation is almost equivalent to eq. (1), except for the dynamic scaling factors
αE (t ) = E E (t ) − v(t ) and αI (t ) = E I − v(t ). For biologically plausible neuron parameters
v(t ) ¿ E E , so one might argue that αE (t ) is almost constant. However, for E I , the difference

between the reset or resting potential—if any—is just a few millivolts, causing a strong
interdependence between the inhibitory current and v(t ) [7].
These potential deviations for inhibitory synapses are supported by the experiments

8

λI = 1000 s−1

λI = 600 s−1

λI = 800 s−1

λI = 400 s−1

λI = 200 s−1

a cond

λI = 0 s−1

a cur

400
350

Output rate [s−1 ]

300
250
200
150
100
50
0
0

200

400
600
Excitatory rate λE [s−1 ]

800

1000

(a) Response curves
λI = 1000 s−1

λI = 600 s−1

Output rate difference a cond − a cur [s−1 ]

λI = 800 s−1

λI = 400 s−1

λI = 200 s−1
λI = 0 s−1

0

−20
−40
−60
−80

0

200

400
600
Excitatory rate λE [s−1 ]

800

1000

(b) Response curve difference
Figure 4: Neuron response curves a cur (λE ), a cond (λE ) of a LIF neuron with current-based or
conductance-based synapses under varying inhibitory bias currents J bias . The excitatory input
spike train is Poisson-distributed with rate λE over 10 s. Lines represent the mean over a thousand
trials, shaded areas the standard deviation.

9

λI = 1000 s−1

λI = 600 s−1

λI = 800 s−1

λI = 400 s−1

λI = 200 s−1

a cond

λI = 0 s−1

a cur

400
350

Output rate [s−1 ]

300
250
200
150
100
50
0
0

200

400
600
Excitatory rate λE [s−1 ]

800

1000

(a) Response curve
λI = 1000 s−1

λI = 600 s−1

Output rate difference a cond − a cur [s−1 ]

λI = 800 s−1

λI = 400 s−1

λI = 200 s−1
λI = 0 s−1

0

−20
−40
−60
−80

0

200

400
600
Excitatory rate λE [s−1 ]

800

1000

(b) Response curve difference
Figure 5: Neuron response curves a cur (λE ), a cond (λE ) of a LIF neuron with current-based or
conductance-based synapses for an input spike train arriving at an inhibitory synapse. Both, the
excitatory and the inhibitory input spike trains are Poisson-distributed with rates λE , λI over 10 s.
Lines represent the mean over a thousand trials, shaded areas the standard deviation.

10

J bias = −30.0 nA

J bias = −18.0 nA

Average membrane current 〈J (t )〉 [nA]

J bias = −24.0 nA

J bias = −12.0 nA

J bias = −6.0 nA

J cond
J cur

J bias = 0.0 nA

30
20
10
0

−10
−20
−30
0

200

400
600
Excitatory rate λE [s−1 ]

800

1000

(a) Average membrane current for inhibitory J bias
λI = 1000 s−1

Average membrane current 〈J (t )〉 [nA]

λI = 800 s−1

λI = 600 s−1
λI = 400 s−1

λI = 200 s−1
λI = 0 s−1

J cond
J cur

30
20
10
0

−10
−20
−30
0

200

400
600
Excitatory rate λE [s−1 ]

800

1000

(b) Average membrane current for inhibitory input spikes.
Figure 6: Average membrane current J of a LIF neuron with conductance-based synapses for an
input spike train arriving at an inhibitory synapse. Both, the excitatory and the inhibitory input
spike trains are Poisson-distributed with rates λE , λI over 10 s. Lines represent the mean over a
thousand trials, shaded areas the standard deviation.

11

Table 1: Neuron and synapse parameters used throughout this report unless specified otherwise.
“Normalized values” refers to the scaled LIF model commonly used in neuron simulators such
as Nengo. Parameters for the conductance-based synapses are scaled according to the linear
transition model in section 5, eq. (11). Synaptic filters are assumed to be normalized to unit area –
whenever this is not the case divide the synaptic weights by τsyn for the denormalized values.
Neuron and synapse parameters
Neuron parameters
Parameter

Value

Membrane capacitance
Membrane time constant
Leak conductivity
Resting potential
Reset potential
Threshold potential
Refractory period

CM
τRC
gL
EL
v reset
v th
τref

1.00
20.00
50.00
−65.00
−65.00
50.00
2.00

Normalized value

nF
ms
nS
mV
mV
mV
ms

1.00
20.00
50.00
0.00
0.00
1.00
2.00

ms
s−1

ms

Current-based synapses
Parameter

Synaptic time constant
Excitatory weight
Inhibitory weight

Value

τsyn
w+
w−

5.00
35.00
−35.00

Normalized value

ms
pA
pA

5.00
2.33
−2.33

ms
10−9
10−9

Conductance-based synapses
Parameter

Synaptic time constant
Excitatory reversal potential
Inhibitory reversal potential
Excitatory weight
Inhibitory weight

Value

τsyn
EE
EI
w+
w−

5.00
0.00
−80.00
0.61
1.56

Normalized value

ms
mV
mV
nS
nS

5.00
4.33
−1.00
40.60
103.70

ms

10−9
10−9

presented in figs. 4 and 5. In each individual experiment, a single LIF neuron with either
current or conductance-based synapses is simulated (see table 1). Parameters for the
conductance-based synapses are matched to the current-based synapses with the linear
transition model discussed in section 5, eq. (11), so the neuron should behave similarly
in both cases. The neuron receives an excitatory input spike train with Poisson statistics
and rate λE . In fig. 4 an inhibitory bias current J bias is injected into the neuron, whereas
in fig. 5 the neuron receives spikes at an inhibitory synapse with rate λI . The inhibitory
currents/rates were matched such that the average neuron rate surpasses zero for approximately the same excitatory input (the x-intercepts are in similar locations). Strikingly,
for excitatory input only (J bias = 0, λI = 0), there are only minor deviations between the

neuron response curves. For an inhibitory J bias , the neuron with conductance-based
synapses has a slightly smaller output rate than the neuron with current-based synapses,

12

though this difference converges to zero for high output rates. However, if the inhibitory
bias current is replaced by an inhibitory synaptic input, the deviations from the neuron
with current-based synapses are much stronger: the output rate is significantly reduced
in the presence of inhibitory synaptic input, especially for large input spike rates. Figure 6
shows the average current flowing into the membrane in the above experiments. For
constant inhibitory bias current, the current increases almost linearly with the excitatory spike rate, whereas inhibitory spikes cause a saturation of the input current. This
non-linearity is likely caused by inhibitory spikes injecting a larger negative current in the
presence of excitatory input due to the increased average membrane potential.

5 Average membrane potential estimation
There are at least two possible approaches regarding the integration of conductance-based
synapses into the NEF. A “natural” approach would be to measure the actual neuron
response curves A as required for the calculation of decoders in eq. (3). However, the
neuron response is no longer defined in terms of a current J but as a two-dimensional
function of g E and g I , which slightly complicates matters, and is not the focus of this
report.
Alternatively, we can try to transform a network of neurons with current-based
synapses into networks of neurons with conductance-based synapses. Again, this assumes that the switch to conductance-based synapses does not have a strong effect on
the neuronal dynamics. For this approach, we calculate weights W as if the network was
still using current-based synapses, but rescale the decomposed W + , W − linearly to account for αE (t ) and αI (t ). Instead of treating these factors as functions time, we estimate
expected values 〈αE (t )〉t and 〈αI (t )〉t . This can be achieved by replacing v(t ) with 〈v(t )〉t .
Clearly, a “one size fits all” average membrane potential does not exist. However, in the
NEF neurons are often in either a mode of (semi) regular spiking or a do not spike at all
for longer periods of time.
A naïve estimate for 〈v(t )〉t in the spiking mode is to assume a linear transition be-

tween v reset and v th . In this case, the average membrane potential is
〈v(t )〉t =

v reset + v th
.
2

(Linear transition)

(11)

However, the membrane potential transition for a neuron is not linear. The force driving
the membrane potential towards E L gets larger with higher v(t ), whereas the force driving
towards E E gets smaller (for conductance-based neurons). Correspondingly, and as
illustrated in fig. 7, v(t ) progresses more slowly through high voltage regimes, causing the
actual 〈v(t )〉t to be larger than the prediction made by the linear model.
13

Membrane potential v(t ) [mV]

Conductance based

Current based

Linear transition

−50
−52
−54
−56
−58
−60
−62
−64
−66

0

1

2

3

4

5

6

7

8

0

Time to spike t [ms]

5

10

15

20

25

Time to spike t [ms]

Figure 7: Idealized membrane potential traces. Comparison of the membrane potential transition
from v reset = −65 mV to v th = −50 mV for a LIF neuron with conductance-based synapses, a LIF
neuron with current-based synapses, and a simple linear model. The left and right graphs show
different transition times (9 ms and 27 ms). Conductances/currents were assumed to be constant.
Dotted lines show the average for each of the three models.

Under the assumption of fast enough regular input spikes we can derive a better approximation for the expected value. For high input rates g E (t ) and g I (t ) are approximately
constant. Specifically, for an infinite input spike train a(t ) with constant inter-spike interval ∆t and a normalized exponential low pass filter h(t 0 ) with time constant τ, it holds
(see appendix A for the derivation)
〈g (t )〉t = 〈(a ∗ h)(t )〉t =

1
.
∆t

For constant conductance, eq. (6) can be written as
Cm

dv(t )
= g L · (e L − v(t )) + g E · (e E − v(t )) + g I · (e I − v(t )) + J bias
dt
= −v(t ) · (g L + g E + g I ) + g L e L + g E e E + g I e I + J bias ,

which is brought into a fairly general form by substituting
λ=

gL + gE + gI
,
Cm

µ=

e L g L + e E g E + e I g I + J bias
.
Cm

14

The closed-form solution for an initial value of v(0) = v 0 is
dv(t )
= −λv(t ) + µ ,
dt

³
µ ´ −λt µ
v(t ) = v 0 −
·e
+ .
λ
λ

If the synaptic current suffices to generate output spikes, the time-to-spike t th after which
the threshold potential v th is reached is given as
Ã
µ!
v th − λ
1
t th = − log
µ .
λ
v0 − λ

(12)

The average membrane potential in the period between two spikes can now be calculated
by setting v 0 = v reset , integrating over the above equation, and dividing by t th :
1
〈v(t )〉t =
t th

t th

Z
0

v(t ) dt =

µ
v0 − λ ³

λt th

´ µ
1 − e −λtth +
λ

For g E → ∞ the expected value 〈v(t )〉t is
〈v(t )〉t =

log

³

E E −v th
E E −E L

log

´
³

· E E − E L + v th
´
.

E E −v th
E E −E L

(Conductance-based)

(13)

Explicitly, these calculations do not include the refractory period τref . The LIF neuron
model assumes all currents flowing into the cell membrane to be zero during the refractory
state. As a consequence, conductance-based synapses do not change the behavior of
the model during the refractory period, so the average membrane potential calculation
should be independent of τref .
To give a sense for the difference between the two estimates discussed so far: for
the neuron parameters listed in table 1 the linear transition model eq. (11) estimates an
average value of −57.5 mV, whereas the conductance-based transition model estimates a
slightly higher value of −57.17 mV for an infinitely strong excitatory input.

Figure 8 shows numerical measurements of the average membrane potential for the

experimental setup presented in the previous section. For a constant bias current J bias and
no inhibitory spikes, the membrane potential converges to the estimate from eq. (13) as
λE increases. In the presence of inhibitory input spikes, 〈v(t )〉t is consistently larger than
in the J bias experiment and reaches values clearly above the estimate. Unintuitively, and

despite output-rate deviations being higher (figs. 4 and 5), the mean error between the estimated average membrane potential and the numerically measured value is consistently
smaller when using inhibitory spikes instead of a bias current J bias . This is especially true
for high λI in comparison to high J bias (i.e. 5.5 mV RMSE for λI = 1000 s−1 , 6.5 mV RMSE

for J bias = −30 nA). Consequently, it might be worthwhile to investigate the effect of not
15

J bias = −30.0 nA

J bias = −12.0 nA

Numerical

J bias = −18.0 nA

J bias = 0.0 nA

Linear estimate

J bias = −6.0 nA

Estimate

Average membrane potential 〈v(t )〉 [mV]

J bias = −24.0 nA

−54
−56
−58
−60
−62
−64
0

200

400
600
Excitatory rate λE [s−1 ]

800

1000

(a) Inhibitory J bias
λI = 1000 s−1
λI = 800 s−1
λI = 600 s−1

λI = 400 s−1

Numerical

λI = 0 s−1

Linear estimate

λI = 200 s−1

Estimate

Average membrane potential 〈v(t )〉 [mV]

−52
−54
−56
−58
−60
−62
−64
0

200

400
600
Excitatory rate λE [s−1 ]

800

1000

(b) Inhibitory input spikes
Figure 8: Estimate and measured average membrane potentials for varying excitatory spike rates
λE and either a constant bias current J bias or an inhibitory spike train with rate λI . Lines represent
the mean over a thousand trials, shaded areas the standard deviation.

16

using an external current source J bias in the NEF altogether without negatively affecting
the variety of tuning curves. The next section contains a simple technique achieving
exactly this by decoding J bias from the pre-population.

6 Elimination of the bias current
As pointed out in eq. (2), the NEF generally defines the current injected into the membrane
J (t ) of a neuron j as J (t ) = J bias + J syn (t ). For a population of n neurons, this can be written

in matrix notation

~
J (t ) = ~
J bias + diag(~
α) · E D · (~
a ∗ h)(t ) = ~
J bias + W · (~
a ∗ h)(t ) ,
where ~
α are n-dimensional vectors, E is the n × d encoding matrix (d being the
J bias and ~

dimensionality of the vector represented by the population), D is d × m decoding matrix

a ∗h)(t ) is the
(m being the size of the pre-population), W is the n ×m weight matrix, and (~

filtered, m dimensional activity of the pre-population. We can approximate this equation
as
¡
¢
~
J (t ) ≈ D 0 + diag(~
α) · E D · (~
a ∗ h)(t ) = W 0 · (~
a ∗ h)(t ) ,

where D 0 is a n ×m matrix decoding the vector ~
J bias from the pre-population. Analogously
to the L 2 approximation in eq. (3), we can approximate D 0 as

¡ bias
¢
¡
¢−1
,
Y0= ~
J
, . . . ,~
J bias = D 0 A ⇒ D 0 = Y 0 A T A A T + I σ2
{z
}
|
m times

where Y 0 is the n × N dimensional target matrix (with N being the number of evaluation

points), and A the n × N neuron response matrix. Using the adapted weight matrix

W 0 = D 0 + W instead of J bias should not significantly alter the behaviour of the system. As
discussed in section 4, we can split W 0 into excitatory weights W 0+ and inhibitory weights

W 0− for use with conductance-based synapses.

7 Benchmark experiments
In this section we present two benchmark networks: a feed-forward communication channel, and an integrator. The first network tests how well one can decode a function from
a conductance-based synapse population with respect to pure feed-forward processing
and a constant stream of input. The second network assess how well dynamical systems
are realized. We then analyze a potential cause of unexpected behaviour in the recurrent
17

network by empirically measuring neuron tuning curves.
Communication channel experiments The communication channel experiment consists of a chain of ` neuron ensembles with one hundred neurons each. All connections
compute the identity function. The input to the system is a bandwidth limited white noise
signal. As a benchmark measure, we compare the decoded output of the last ensemble in
the chain to the input value. The experiment is executed for standard LIF neurons, LIF
neurons with conductance-based synapses and eliminated J bias (see section 6), as well
as LIF neurons with conductance-based synapses and active J bias . Results are shown in
fig. 9.
As is clearly visible, there is no significant difference in error for the three neuron types
for short communication channels. Notably however, across all experiments, currentbased synapses are always best for inputs with low bandwidth, and using J bias in conjunction with conductance-based synapses increases the error. Apart from this, conductancebased synapses are rather competitive compared to current-based synapses, and as the
bandwidth increases, the error is mostly dominated by the baseline error obtained when
filtering the input signal with the synaptic filter. Interestingly, current-based synapses are
only better in high bandwidth scenarios when increasing the communication channel
length to eight. At least the error caused by J bias can be explained with the observations
laid out in section 5. Specifically, the average membrane potential deviates from the
estimated values for a large input spike rate range.
Integrator experiments In the integrator experiment, a standard NEF integrator with
100 ms synaptic time constant for the recurrent connection is tested. A value x ∈ [0, 1] is

fed into the integrator for one second, theoretically causing the integrator to converge
to x. The simulation is continued for nine seconds and the difference between x and
the actual value is recorded. Figure 10 shows the results over varying integrator neuron
counts. For standard current-based synapses, the error is relatively small and converges
to zero with increasing neuron count. The integrator with conductance-based synapses
exhibits much larger errors. The error does not significantly decrease with increasing
neuron count and is significantly higher when not eliminating J bias . When recording
the integrator value trace over time, the integrator often converges to significantly larger
than desired values. This is shown in fig. 10b, which depicts the average final integrator

value. While the integrator with current-based synapses on average converges to the value
0.5, as expected when sampling uniformly from the interval [0, 1], the integrator with
conductance-based synapses converges to significantly larger values.

18

Cur.

Cond. (with J bias )

Cond.

Ref.

0.4

RMSE

0.3
0.2
0.1
0.0
10

20
30
Input bandwidth [Hz]

40

50

40

50

40

50

(a) n = 2
0.4

RMSE

0.3
0.2
0.1
0.0
10

20
30
Input bandwidth [Hz]

(b) n = 4
0.4

RMSE

0.3
0.2
0.1
0.0
10

20
30
Input bandwidth [Hz]

(c) n = 8
Figure 9: Communication channel benchmark. A bandwidth limited white noise signal is fed into
a series of neuron ensembles (where the number of ensembles n is the “communication channel
length”), with a synaptic time constant of 5 ms for all connections. The graphs show the RMSE
between input and output signal averaged over 16 runs. The dashed line depicts the baseline error
obtained when filtering the input n times with the synaptic filter.

19

Cur.

Cond.

Cond. (with J bias )

0.5

RMSE

0.4
0.3
0.2
0.1
0.0
102

103
Number of neurons

(a) Integration error

(b) Average integrator value
Figure 10: Results for the integrator benchmark. (a) Shows the average integrator representation
error over 24 runs in which the integrator is driven to values in the interval [0, 1]. Error bars
correspond to the standard deviation. (b) Shows the average value that the integrator converges to
over all 24 runs. The dashed line depicts the expected value of 0.5.

Empirical tuning curve measurment Figure 11a shows empirically measured tuning
curves for a LIF population with either current or conductance-based synapses. Tuning curves are measured by feeding a bandwidth limited (5 Hz) white noise signal x(t )
into a standard current-based LIF ensemble with 1000 neurons connected to the target
population with 100 neurons. The neural activities a(x(t )) are recorded, and the inverse
mapping x(a) is calculated. The measured tuning curves are similar, however, as shown
in fig. 11b, they mainly differ for low output rates—the output rates of conductance-based
neurons systematically are too small. When simulating dynamical systems such as an

20

Current based

400

Conductance based

350

Output rate s−1

300
250
200
150
100
50
0
−1.0

−0.5

0.0

0.5

1.0

Input value x

−1.0

0.0

−0.5

0.5

1.0

Input value x

(a) Tuning curves

10

1

Cond. vs. ground truth

Cond. vs. cur.

Cur. vs. ground truth

RMSE

100

10−1

10−2

0

50

100

150

200

Output rate s

250

300

350

−1

(b) Error over output rate
Figure 11: Tuning curve experiment results. (a) Empirically measured tuning curves for both
current and conductance-based synapses. (b) RMSE between the tuning curves for varying output
rate regimes. Ground truth refers to the non-spiking LIF firing rate model. In the “cond. vs. cur.”
graph, the empirical measurements for the current-based model are used as a reference.

integrator, the imbalance caused be the later spike onset (which prevents both excitation
and inhibition) may be a reason for the systematic tendency to over-estimate the integral
of the signal.

21

Summary In the feed-forward case, the simplistic transformation presented in section 4
allows neuron ensembles with conductance-based synapses to function as well as their
current-based counterparts, given that J bias is eliminated from the neuron model (as
outlined in section 6). For mostly unknown reasons, the performance of the conductancebased synapses breaks down once recurrent networks are implemented. This may be
caused by delayed spike onset in the tuning curves not accounted for when computing
decoders.

8 Conclusion
In this report, we described both the NEF and LIF neurons with conductance-based
synapses. We presented a simple transformation translating existing current-based NEF
networks into their conductance-based counterparts. A sequence of benchmark experiments shows that this transformation yields relatively little error for feed-forward networks. However, when implementing dynamics such as integrators there are systematic
deviations from the desired behaviour. Experiments show that the neuron tuning curves
differ significantly between neurons with current-based synapses and the transformed
conductance-based version. They exhibit a delayed spike onset, which is a potential
explanation for the observed deviations in recurrent networks.
Unfortunately, this problem can likely not be solved with a simple transformation of
pre-computed weight matrices. Instead, the two-dimensional neuron response curve
of a neuron with excitatory and inhibitory conductance-based synapses must be taken
into account. While the neuron response curve can be described analytically for constant,
noiseless input, the non-negativity of weight matrices W + and W − mandates an iterative
optimization scheme. Preliminary research indicates that these methods are feasible, yet
computationally more expensive than standard NEF solvers, especially because the entire
weight matrix W is optimized and not the significantly smaller and reusable decoder D.

References
[1]

Chris Eliasmith and Charles H. Anderson. Neural Engineering: Computation, Representation, and Dynamics in Neurobiological Systems. Cambridge, Massachusetts:
MIT press, 2003. 380 pp.

[2]

Christof Koch. Biophysics of Computation: Information Processing in Single Neurons.
Oxford, United Kingdom: Oxford University Press, 1999. 562 pp.

[3]

Wulfram Gerstner and Werner M. Kistler. Spiking Neuron Models: Single Neurons,
Populations, Plasticity. LCCB: 2002067657. Cambridge University Press, 2002.
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[4]

Alan L. Hodgkin and Andrew F. Huxley. “A Quantitative Description of Membrane
Current and Its Application to Conduction and Excitation in Nerve”. In: The Journal
of Physiology 117.4 (1952), pp. 500–544.

[5]

J. Schemmel et al. “A Wafer-Scale Neuromorphic Hardware System for Large-Scale
Neural Modeling”. In: Proceedings of 2010 IEEE International Symposium on Circuits
and Systems. May 2010, pp. 1947–1950.

[6]

Thomas Pfeil et al. “Six Networks on a Universal Neuromorphic Computing Substrate”. In: Frontiers in Neuroscience 7 (2013), p. 11.

[7]

Arnd Roth and Mark C. W. van Rossum. “Modeling Synapses”. In: Computational
Modeling Methods for Neuroscientists. Ed. by Erik De Schutter. The MIT Press, 2009,
pp. 139–159.

[8]

Heinrich Reichert. Neurobiologie. 2., neubearb. und erw. Aufl. Stuttgart [u.a.]: Thieme,
2000. VII, 250 S. : Ill., graph. Darst.

[9]

E. Kandel et al. Principles of Neural Science. 5th ed. McGraw-Hill Education, 2012.

[10]

Wulfram Gerstner et al. Neuronal Dynamics: From Single Neurons to Networks and
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[11]

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(2008), pp. 1473–1494.

23

Filtered value

12

13

10

12

8

11

6

10

4

9

2

8

0

7
0

1

2

3

4

5

4.70 4.75 4.80 4.85 4.90 4.95 5.00

Time [s]

Time [s]

Figure 12: Example of a low-pass filtered regular spike train. In this example, individual input
spikes are ∆t = 0.1 apart. The filter time constant is τ = 0.3. The right graph is a magnification
of the left one. Dotted lines correspond to ĥ and ĥ − τ−1 , the dashed line to the final average
∆t −1 = 10.

A Expected value of a filtered, regularly spaced spike train
Claim Given an infinite spike train with regular spacing a(t ) =
1
τ

normalized exponential low-pass filter h(t ) = · e
〈h ∗ a〉 = lim

Z

t

t →∞ 0

−t /τ

it holds

P∞

i =0 δ(t

− ∆t · i ) and a

(h ∗ a)(t 0 ) 0
1
dt =
.
t
∆t

Proof This directly follows from linear systems theory and the unit-gain of the exponential low-pass. Alternatively, the supremum ĥ of h ∗ a can be expressed in terms of a
geometric series

∆t

∞
∆t·`
1
1 X
eτ
ĥ = sup(h ∗ a) = ·
e − τ = · ∆t
.
τ `=0
τ e τ −1

Correspondingly, and as illustrated in fig. 12, after a finite amount of time the filtered spike
train approximately oscillates between ĥ and ĥ−τ−1 every ∆t. The function between these
two points is an infinite sum of exponentials, which itself is an exponential g (t ) = a · e −bt ,
!

!

with g (0) = ĥ and g (∆t) = ĥ − τ−1 . Solving for a, b yields a = ĥ and b = τ−1 . The average of
g (t ) over [0, ∆t) is

∆t
Z ∆t
t
1
eτ
· ∆t
·
e − τ dt
∆t · τ e τ − 1 0
0
∆t
∆t
∆t
³
´
1
1
eτ
eτ
e τ −1
1
− ∆t
τ
=
· ∆t
· 1−e
=
· ∆t
·
=
. 
∆t
∆t e τ − 1
∆t e τ − 1
∆t
eτ

〈h ∗ a〉 = 〈g (t )〉 =

1
·
∆t

Z

∆t

g (t ) dt =

24

