A Study of Sub-threshold Resonant Dendrites under Periodic Stimuli
by Jamie Luo
Supervised by Yulia Timofeeva
Abstract
Dendrites form a major part of the structure and function of neurons receiving and transmitting numerous
stimuli from the synapses to the soma. The model described here incorporates a quasi-active membrane into
the dendritic tree. This feature causes resonance effects which are not observed in the passive membrane
model. In earlier work by Coombes et al. (2007) Laplace transform theory was applied to construct a transfer
function for an arbitrary branching structure. We use this method to run simulations investigating the
response of an infinite branch to the sine and chirp stimuli. In addition to confirming the resonance effect
observed in experiments we also discovered a frequency dependent, phase-shifting phenomenon. Further
analysis involving methods from steady-state circuit theory are applied to derive an exact analytical prediction
for the steady-state response to a sinusoidal input. This approach is then extended onto an arbitrary branching
structure with some constraints. Implications for the response to a chirp stimulus are also discussed.
Introduction
The dendrites of neurons often form complicated branching structures. They conduct charge from
the synapses to the soma, eventually bringing the soma to a threshold value where it fires.
Understanding the response of these structures to different stimuli is a means to us gaining insight
into how neurons operate.
This investigation focuses on the sub-threshold dynamics of dendrites and their responses to
periodic stimuli in the form of the sine and the chirp stimuli (example stimuli depicted in Fig. 1). The
focus on the chirp input is inspired by the experimental work of Ulrich (2002) and Narayanan and
Johnston (2008). Many neurons exhibit an amplified voltage response when stimulated at
preferential frequencies. The chirp stimulus is essentially a sine stimulus whose frequency increases
linearly with time. The resonance effect causes a peak in the chirp’s voltage response wave at a
frequency which the experimentalists mentioned then ascribed as being the resonant frequency of
the dendrite as illustrated in Fig. 2.
Much of the earlier work on modelling dendritic structures was originally conducted assuming a
passive membrane which only comprised of an RC circuit, which represents some passive leakage (R)
of charge and the phospholipid bilayer (C). This setup fails to take into account the dynamics induced
by the nonlinear voltage-gated ion channels. It was mathematically demonstrated by Mauro et al.
(1970) that one could linearize such channel kinetics about some steady-state and still accurately
describe the resonance effects. From a circuit theory perspective the linearization includes
inductances in parallel to the RC circuit generating an RLC circuit as depicted in Fig. 3. Koch called the
resulting membrane model quasi-active to distinguish it from the passive (RC) and fully non-linear
cases. The Laplace (frequency) domain solution for an arbitrary branching structure was later
computed by Koch and Poggio (2005), while work by Abbott et al. (1991) showed how to compute
the response functions for a passive branched dendrite explicitly in the time domain. This method
was extended to the case of the quasi-active membrane by Coombes et al. (2007). We exploit this
formalism to perform simulations and analysis first on the infinite branch and then on other
branching structures.
1
In addition to the resonance effect we observed that there was a phase-shifting phenomenon that
was dependent on frequency and space (Fig. 7). We employ steady state methods from AC analysis
in electric circuit theory to derive a method for finding this phase-shift and the magnitude of the
response induced by a sine stimulus for the infinite branch. We conclude with a discussion on
applying these results to predicting the response to the chirp stimulus and extending the approach
to an arbitrary branching structure.
Chirp Stimulus
Sine Stimulus
1
sin((f0+( /2)t)t)
1
sin(t)
0.5
0
-0.5
-1
0
2
4
6
8
0
-0.5
-1
10
Time t
0
2
4
6
Time t
sin( 𝑓0 +
sin(𝜔𝑡)
Fig. 1
0.5
𝜔
𝑡 𝑡)
2
Examples of a sine (left) and a chirp (right) stimulus.
Fig. 2 Extract from Ulrich (2002) where a chirp
was injected in to a rat neocortical pyramidal cell
via the dendritic patch electrode and the resulting
voltage response was recorded by the somatic
electrode. Notice the peak indicated around a
preferential frequency. This is the resonance effect
of the dendritic membrane.
2
8
10
Model Outline
We shall outline the model and the Green’s function method for finding the solution. Full details can
be found in Coombes et al. (2007).
Quasi-active membrane
Traditionally most neuron models have used a purely passive membrane component modelled by an
RC circuit (illustrated below in Fig. 3), where the resistance and capacitance are in parallel. Active
membrane models typically expand upon the framework of the passive membrane incorporating ion
channels in the form of active (non-linear) elements into their membrane models in parallel with the
RC components (Fig. 3).
Let us begin with a nonlinear ionic membrane current, 𝐼 = 𝐼(𝑉, 𝑤1 , … , 𝑤𝑁 ) with voltage 𝑉 and gating
variables, 𝑤𝑘 satisfying, 𝜏𝑘 𝑉 𝑤𝑘 = 𝑤𝑘,∞ 𝑉 − 𝑤𝑘 for all 1 ≤ 𝑘 ≤ 𝑁. Then we linearize around a
steady state (ss), (𝑉𝑠𝑠 , 𝑤1,∞ (𝑉𝑠𝑠 ), … , 𝑤𝑁,∞ (𝑉𝑠𝑠 )) for the active membrane which leads us to replace
the active elements with 𝑁 inductance and resistance terms producing the so called quasi-active
membrane model.
The inclusion of inductance terms mean the quasi-active membrane is represented by an RLC circuit
with the inductance terms in parallel to the resistance and capacitance. For a general ionic current
every type of ion channel corresponds to an active term in parallel with the RC circuit and in our
linearization each of these is replaced by an inductance in series with a resistance (Fig. 3).
From a general current balance equation, 𝐶
𝑑𝑉
𝑑𝑡
= −𝑔𝐿 𝑉 − 𝑉𝐿 − 𝐼 + 𝐼𝑖𝑛𝑗 the linearized equations
are derived to be
𝑑𝑉
𝑉
𝐶
=− −
𝑑𝑡
𝑅
𝐿𝑘
𝑁
𝐼𝑘 + 𝐼𝑖𝑛𝑗 ,
𝑘=1
𝑑𝐼𝑘
= −𝑟𝑘 𝐼𝑘 + 𝑉 ,
𝑑𝑡
where
𝑅 −1 = 𝑔𝐿 +
R
C
R
𝜕𝐼
𝜕𝑉
𝑎𝑡 𝑠𝑠
gL
C
.
r
R
C
L
Fig. 3 From left to right we have the circuit diagrams of
the passive (RC), active and quasi-active (RLC) membranes
3
Koch’s general infinite branch framework
Let us first recap the general model for the infinite branch employed by Koch (1984). The neuron
comprises of its membrane coupled to a spatial axis giving us the model depicted below in Fig. 4 in
terms of impedances (defined on p.9). Our model is a specific case where 𝑧𝑚 is the quasi-active
membrane described earlier coupled with a purely resistive term for the longitudinal impedance 𝑧𝑎 .
𝑧𝑎
𝑧𝑎
𝑧𝑚
𝑧𝑚
𝑧𝑚
Ra
r
C
R
Ra
R
r
C
R
L
r
C
L
L
Fig. 4 Above is Koch’s generalised infinite branch
model in term of impedances. Below that is our
particular model which is a special case of Koch’s model.
Laplace Transform Method for solving Differential Equations
The Laplace Transform (L.T.) of a function, 𝑓(𝑡) is defined to be 𝐹(𝑠) =
∞
0
𝑑𝑡𝑒 −𝑠𝑡 𝑓(𝑡). The L.T. can
be applied to solve differential equations as illustrated below.
Input Signal
CIRCUIT
Output Signal
(Differential Equation)
Fig. 5
L.T. of
Laplace transform of
L.T. of
Input Signal
CIRCUIT
Output Signal
Simple diagrammatic explanation of the Laplace Transform method we will use.
4
Single Infinite Branch
The main assumption of cable theory is that the only axis of relevance is the longitudinal one,
reducing the degrees of freedom from three to one. On a single infinite branch, where 𝑉 = 𝑉(𝑋, 𝑡),
the system of differential equations governing our system is derived from coupling the standard
cable equation 𝜕𝑉/𝜕𝑡 = −𝑉/𝜏 + 𝐷𝜕 2 𝑉/𝜕𝑋 2 to resonant currents yielding
𝜕𝑉
𝑉
𝜕2 𝑉 1
=− +𝐷 2−
𝜕𝑡
𝜏
𝜕𝑋
𝐶
𝐿𝑘
𝑁
𝐼𝑘 − 𝐼𝑖𝑛𝑗 ,
𝑘=1
𝑑𝐼𝑘
= −𝑟𝑘 𝐼𝑘 + 𝑉.
𝑑𝑡
Taking the Laplace transform of the system and rescaling space by 𝑥 = 𝛾(𝜔)𝑋, where
𝛾2 𝜔 =
1 1
1
+𝜔+
𝐷 𝜏
𝐶
𝑘
1
,
𝑟𝑘 + 𝜔𝐿𝑘
simplifies the system to (with vanishing initial conditions)
−𝑉𝑥𝑥 𝑥, 𝜔 + 𝛾 2 𝜔 𝑉 𝑥, 𝜔 = 𝐴 ,
𝐴 𝑥, 𝜔 =
1
𝑥
𝐼𝑖𝑛𝑗
,𝜔 .
2
𝐶𝐷𝛾 𝜔
𝛾 𝜔
The Green’s function of the operator (1 − 𝑑𝑥𝑥 ) is 𝐻∞ 𝑥 =
𝑒 −|𝑥 |
2
, implying that the frequency
domain solution is
∞
𝑉 𝑥, 𝜔 =
−∞
𝑑𝑦𝐻∞ 𝑥 − 𝑦 𝐴 𝑦, 𝜔 ,
or in the original co-ordinates
∞
𝑉(𝑋, 𝜔) =
−∞
𝑑𝑌𝐺∞ 𝑋 − 𝑌, 𝜔 𝐼𝑖𝑛𝑗 𝑌, 𝜔 /𝐶 ,
where
𝐺∞ 𝑋, 𝜔 =
𝐻∞ 𝛾(𝜔)𝑋
.
2𝐷𝛾(𝜔)
Taking the inverse L.T. will give the solution in the time domain
𝑡
𝑉 𝑋, 𝑡 =
∞
𝑑𝑠
0
−∞
𝑑𝑌𝐺∞ 𝑋 − 𝑌, 𝑡 − 𝑠 𝐼 𝑌, 𝑠 .
Typically though, a closed form solution does not exist here requiring a numerical inversion.
Importantly one sees that the required convolution to obtain a time domain solution is only a
multiplication in the frequency domain greatly simplifying the difficulty of simulations.
5
Simulations
Using the framework described above Matlab simulations were run to simulate the response of the
system to periodic stimuli. The main focus of simulations was a single sine or chirp input stimulus on
a single infinite branch which is the equivalent of multiplying the input by a delta function. The fast
Fourier transform (fft) was used as a substitute for the L.T. with the term 𝑠 being replaced by 𝑖𝜔. For
our Green’s function the continuous Fourier transform is equivalent to evaluating the bilateral L.T.
with complex argument 𝑠 = 𝑖𝜔. Furthermore as we only consider causal signals which are defined to
be zero for negative time then they are also equivalent to the unilateral L.T. that we defined earlier.
We can now use the inverse fft function in Matlab as our method for numerical inversion. We
exploited the fact that the convolution which we need to calculate in the time domain is simply a
multiplication in the frequency domain. This makes it possible to simulate any type of input with
ease. It is customary to test if such a method is accurate and to that effect numerical pde solvers
were applied to check that the simulations were accurate.
The model has also been shown to be a good predictor of the behaviour of the non-linear ionic
current for step function inputs (Coombes et al. (2007)). For periodic inputs, Fig. 6 below depicts a
comparison of the voltage response to an injected sine current in our linear model to the fully nonlinear system using Neuron to perform numerical simulations of both.
V (mV)
t (ms)
Fig. 6
Numerical NEURON simulations on the infinite cable
with the RLC circuit model and the nonlinear 𝐼𝑕 current. The
good fit suggests that our model is a valid linearization of the
active one. The input current was 𝐼_𝑖𝑛𝑗 (𝑡) = −0.3𝑠𝑖𝑛(0.07𝑡).
The branch parameters were 𝐶 = 1 𝜇𝐹/𝑐𝑚2 , 𝜏 = 20 𝑚𝑠 and
𝐷 = 50,000 𝜇𝑚2 /𝑚𝑠. For the RLC circuit the membrane
parameters were 𝑟 = 13,500 Ω 𝑐𝑚2 and 𝐿 = 1,150 𝐻 𝑐𝑚2 .
6
Results and Analysis
Initial Observations of Resonance & Phase-shifting
Initial observations found that the voltage response to a chirp current on the infinite branch shows
resonance at certain frequencies (Fig. 7). This agrees with the experimental work of Ulrich (2002)
and Narayanan and Johnston (2008) mentioned earlier.
These simulations also yielded another phenomena relating to the relative phase-shift between the
responses 𝑉(𝑋, 𝑡) at different spatial locations 𝑋. As can be seen in Fig. 7 the response induced at
the higher frequency end of the chirp stimulus creates a progressive forward shift in the phases of
the voltage responses 𝑉(𝑋, 𝑡) which increases with 𝑋 (distance from the stimulus). Furthermore at
the low frequency end of the chirp we see a backward shift in the phases of the responses
𝑉(𝑋, 𝑡) with increasing 𝑋. This suggested and it was soon observed that there was some critical
frequency 𝜔𝑐 at which there might be phase lock.
In order to investigate the phenomena of resonance and phase shifting the sine stimulus was
analysed and exhibited behaviour similar to that elicited by the chirp. The magnitude of the
response to a sine stimulus was greater around a certain preferential frequency and indeed we see
again the phase shift phenomena (Fig. 8A-C).
V(x,t) for x=0,0.5,1
0.4
x=0
x=0.5
x=1.0
0.3
0.2
V
0.1
0
-0.1
-0.2
-0.3
-0.4
0
2
4
t
6
8
Fig. 7 The voltage responses to a chirp stimulus at three
different points on a single infinite branch. There is a
clear resonance around some preferred frequency and
also phase-shifting associated with frequency. The branch
parameters are 𝑟 = 0.005, 𝐶 = 1, 𝑅 = 1, 𝐿 = 0.01 𝐷 = 1
and the chirp increases frequency at a rate 2.
7
10
𝜔 > 𝜔𝑐
Backward shift in V(x,t) x=0, 0.2, ...,1
A
D
0.3
Voltage Lag
0.2
0.2
V(0,t)
Iinj(0,t)
0.15
0.1
V(x,t) x=0
0.1
V
0
-0.1
0.05
0
-0.05
-0.1
-0.2
-0.15
-0.3
-0.4
-0.2
-0.25
0
1
2
3
4
5
6
7
8
9
0.5
10
1
1.5
2
2.5
t
t
𝜔 < 𝜔𝑐
Forward shift in V(x,t) x=0, 0.2, ...,1
B
E
0.3
Voltage Lead
0.2
0.1
0.1
0.05
V
V(x,t) x=0
0
-0.1
-0.2
0
-0.05
V(0,t)
Iinj(0,t)
-0.3
-0.1
-0.4
0
1
2
3
4
5
6
7
8
9
0.5
10
1
1.5
3
2
F
0.4
1.5
0.3
()=arg(z(i))
0.2
0.1
V
2.5
Impedance plot
No shift in V(x,t) x=0, 0.2, ...,1
0.5
C
2
t
t
0
-0.1
-0.2
RLC lead & lag
RL only lead
RC only lag
1
0.5
0
-0.5
-0.3
-1
-0.4
-0.5
0
1
2
3
4
5
6
7
8
9
-1.5
0
10
t
0.5
1
1.5
2
2.5

3
3.5
4
4.5
5
Fig. 8 A, B & C: voltage response profiles to sine inputs on an infinite branch with membrane
parameters 𝑟 = 0.05, 𝑅 = 1, 𝐶 = 1, 𝐿 = 0.1 (other parameters all set to 1). The inputs were set to the
frequencies 𝜔 = 2, 3.1225 & 5 in plots A, B and C respectively where 𝜔𝑐 = 3.1225. D: an input current
and response at the input location with 𝜔 > 𝜔𝑐 inducing voltage lag. E: an input current and response at
the input location with 𝜔 < 𝜔𝑐 inducing voltage lead. All these plots are intuitively clearer when you
focus on where the peaks of the waves are. F: Plot of the argument of 𝑧(𝑖𝜔) for our RLC circuit, a RC
circuit and a RL circuit. The lag/lead effect in D/E depends on which side of the x-axis this value takes.
There is no lead/lag when 𝜙 = 0, which can only happen for the RLC circuit.
8
RLC circuit & Impedance
As the longitudinal component of the model is purely resistive it is sensible to first consider the
quasi-active membrane in isolation, i.e. a single RLC circuit compartment. The typical means of
analysis of such a circuit relies on looking at the impedance of the circuit. In general impedance is a
complex quantity 𝑧 whereby the polar form conveniently captures both magnitude and phase
characteristics 𝑧 = 𝑧 𝑒 𝑖𝜃 where the magnitude |𝑧| gives the change in voltage amplitude for a given
current amplitude, while the argument 𝜃 gives the phase difference between voltage and current.
One can explicitly write down the impedance of the RLC circuit shown in Fig. 3 (only one inductance
branch) as
𝑧 𝜔 =
(𝑟 + 𝐿𝜔)
.
𝐶𝜔 + 𝑅 −1 𝑟 + 𝐿𝜔 + 1
Let us introduce some basic ideas about impedance and a useful theorem from circuit theory which
will aid us in our analysis of the sine stimulus, which is the equivalent of an AC current.
The impedance 𝑧 satisfies Ohm’s Law: 𝑉 = 𝐼 𝑧 where 𝑉 = 𝑉0 𝑒 𝑖 𝜔𝑡 +𝜙 𝑉 and 𝐼 = 𝐼0 𝑒 𝑖(𝜔𝑡 +𝜙 𝐼 ) are the
complex representations of the voltage and current respectively. The real parts of 𝑉 and 𝐼 yield the
real voltage and current.
Theorem 11:
Let 𝐼(𝑡) be the input to a general linear time-invariant system, and 𝑉 𝑡 be the output, and the
Laplace transform of 𝐼(𝑡) and 𝑉(𝑡) be 𝐼(𝜔) and 𝑉(𝜔)respectively.
Then the output is related to the input by the transfer function 𝐻(𝜔) as
𝑉(𝜔) = 𝐻 𝜔 𝐼 𝜔
In particular if 𝐻(𝜔) is stable, then for any sinusoidal input
𝐼 𝑡 = 𝐴 sin(𝑘𝑡)
the steady-state response is of the form
𝑉 𝑡 = 𝐵 sin(𝑘𝑡 + 𝜙)
where
𝐵 = |𝐻 𝑖𝑘 |
and
𝜙 = arg(𝐻(𝑖𝑘))
We see that as impedance satisfies Ohm’s law it is indeed the transfer function of the RLC circuit.
The above theorem thus applies meaning that we can estimate the phase shift of the response, 𝑉(𝑡)
to the current, 𝐼(𝑡). Note that as we are dealing with a single compartment there is no spatial aspect
to the response 𝑉(𝑡). The RLC circuit exhibits a well understood behaviour in electronic circuit
theory described as the lead or lag of a voltage response to an applied sinusoidal current (Fig. 8D-E).
Applying the theorem we can now plot the argument of the impedance against the frequency of the
input current (Fig. 8F). For 𝜔 > 0, arg(𝑧 𝑖𝜔 ) crosses the 𝜔-axis only once in our membrane model.
This crossing point is the frequency where the lead/lag of the RLC circuit is zero (i.e. the input
1
Adapted from DeCarlo (1995)
9
current and the voltage response are in perfect sync). In comparison the simpler RC circuit (passive
membrane model) has no such crossing point and will only experience lag, while an RL circuit only
exhibits a lead (both are plotted in Fig. 8F for comparison).
It was observed that in simulations of the spatially extended system on an infinite branch that
phase-lock occurred at the critical frequency
𝐿 − 𝐶𝑟 2
.
𝐶𝐿2
This corresponds exactly to the frequency at which the argument for the impedance of the RLC
circuit crossed the 𝑥-axis. This is not a coincidence.
𝜔𝑐 =
Steady-state analysis of the Infinite branch: Phase Shift and Magnitude
To extend this analysis to the entire branch model we simply note that for the infinite branch with a
single input at location 𝑌, the frequency domain solution is simply
𝑉(𝑋, 𝜔) = 𝐺∞ 𝑋 − 𝑌, 𝜔 𝐼𝑖𝑛𝑗 𝑌, 𝜔 /𝐶 .
This implies that the Green’s function 𝐺∞ 𝑋 − 𝑌, 𝜔 /𝐶 we deduced earlier is in fact the transfer
function for this system. It is stable as 𝛾(𝜔), which is the only relevant part of the denominator of
𝐺∞ 𝑋 − 𝑌, 𝜔 /𝐶 only has zeros in the left half of the complex plane implying that this will be where
all the poles of the transfer function are located (this is in fact the definition of stability). Theorem 1
thus applies and so we deduce that for a single sine input of the form,
𝐼 𝑡 = 𝐴 sin(𝑘𝑡)
the steady state response is
𝑉 𝑥, 𝑡 = 𝐵 sin(𝑘𝑡 + 𝜙)
where
𝐵 = |𝐺∞ 𝑋 − 𝑌, 𝑖𝑘 /𝐶 |
and
𝜙 = − 𝑥 𝛾 𝑖𝑘 sin 𝜃(𝑘) − 𝜃(𝑘)
where
𝜃 𝑘 = arg(𝛾(𝑖𝑘))
Focusing on the formula for the phase shift 𝜙 we note that it has a spatial component |𝑥| which
explains why the more distant ones from the input site have a greater the lead/lag of the response
wave. On an infinite branch this also implies a strictly linear relationship between the lead/lag
(shifting) of 𝑉(𝑥, 𝑡) to 𝐼(𝑡) and |𝑥|.
Phase lock occurs for a critical frequency, 𝜔𝑐 = (𝐿 − 𝐶𝑟 2 )/𝐶𝐿2 which corresponds exactly to when
the lead/lag of the single RLC compartment is zero. These coincide because 𝛾 2 𝜔 = 1/(𝐷𝑧 𝜔 ) .
The resonant frequency of the circuit is the frequency 𝑘 for which 𝐵 is maximal. There is only one
maximum in the infinite branch and one can consult Koch (1984) for an in depth analysis.
One can now accurately predict the steady-state response of the infinite cable to any sinusoidal
input. This result has been tested by simulations over varying sets of parameters (an example is
depicted in Fig. 9A).
It is important to point out that the analytical solutions predicted here are only steady state ones. As
the initial potential and current are both set to zero there is an initial transient state which will
10
asymptotically approach the steady-state solutions. We do not analyse this transient state in this
report but it has crucial implications for our discussion of the chirp stimulus. A study of the transient
state is a natural area for further work.
Steady State Fits to V(x,t) for x=0, 0.2, ...,1
Two Branches: Analytical Fits (red) vs Simulations (blue)
B
0.3
A
0.2
0.15
0.2
0.1
0.1
0.05
V
V
0
0
-0.1
-0.05
-0.2
-0.1
-0.3
-0.4
-0.15
-0.2
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
t
t
'Knock-on' effect of a sine input current doubling its frequency
C
D
Attempts to construct a Chirp Envelope
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
V
V
0.1
0
0
-0.1
-0.1
-0.2
-0.2
-0.3
-0.3
-0.4
-0.4
-0.5
2
4
6
8
10
12
14
t
0
5
10
15
20
25
30
35
t
Fig. 9 A: an example of the voltage responses on the infinite branch with the same branch parameters
as in Fig. 8A-C and an input sine wave of frequency 5. The analytical wave forms (green) accurately
predict the steady-state solutions after an initial transient state. B: two branches with different
parameters (branch1: 𝑟 = 4, 𝑅 = 1, 𝐶 = 1, 𝐿 = 10, 𝐷 = 1 and branch 2: 𝑟 = 2, 𝑅 = 1, 𝐶 = 1, 𝐿 = 5, 𝐷 =
1) were simulated and the red lines are analytically predicted waveforms for branch 1 (while the sine
input of frequency 5 was on branch 2 at distance 0.2 from the connecting node) and we see that they are
a good fit. C: the red lines are the predicted magnitude with phase-shift of the chirp response canonically
implied from the sine analysis. As one can see in this case the voltage profile of the simulation is not
accurately replicated. The branch parameters are 𝑟 = 0.5, 𝑅 = 1, 𝐶 = 1, 𝐿 = 1, 𝐷 = 1 and the chirp
increases frequency at a rate 1. D: ‘Knock-on’ effect which complicates the analysis of the chirp. A sine
stimulus that suddenly doubles its frequency from 3 to 6 around 𝑡 = 16 inducing non-trivial transient
behaviour whereby the voltage actually exceeds its predicted steady state amplitude. The branch
parameters are 𝑟 = 0.005, 𝑅 = 1, 𝐶 = 1, 𝐿 = 0.1, 𝐷 = 1.
11
40
Implications for the Chirp stimulus
The original intent was to use the analysis of the sine stimulus to aid us in characterising the
response to the chirp stimulus. We have found and analytically explained both the phenomena of
resonance and phase-shifting with regards to the sine stimulus for its steady-state response. Naively
one would hope to construct an envelope or bounds on the profile of the chirp stimulus. Every point
in time in the injected chirp current corresponds to a particular frequency which is increasing
linearly. Thus for this frequency one can write down the steady-state magnitude and phase-shift of
the response to a sine current of this frequency. Problematically though the chirp response will often
exceed this envelope, and its peak often does not correspond to the peak frequency predicted by
the sine frequency (Fig. 9C). This is possibly due to the fact that the chirp exists permanently in a
transient state as its frequency varies so does the voltage response and correspondingly its phase
shift is not accurately predictable.
Observationally it appears that the peak of the chirp is typically ahead of where the steady state
predicts it to be. This calls into question the appropriateness of using the chirp to determine the
resonant frequency of a dendrite, as our results for the infinite branch suggest this might not
accurately represent the steady-state response of the dendrite.
Extensions to other branching Structures
To generalise the analysis of the sine input onto an arbitrary branching structure one needs to
compute a general transfer function which is shown to be stable. The work of Coombes et al. (2007)
demonstrated the means for deriving the transfer function for an arbitrary branching structure and
so Theorem 1 is applicable so long as the transfer function is shown to be stable.
An Arbitrary branching Structure
First let us quickly summarise the construction of this transfer function. Full details can be found in
Coombes et al. (2007). We keep to the essentials necessary to understand the extension of the
steady-state analysis to an arbitrary tree.
The dendritic tree is composed of finite branches labelled 𝑖 of length 𝑙𝑖 with 0 ≤ 𝑋𝑖 ≤ 𝑙𝑖 defined as
the position on a branch 𝑖. Branches are allowed to meet at nodes or end in terminals and the
dynamics of a branch 𝑖 are governed by the dynamics
𝜕𝑉𝑖
𝑉𝑖
𝜕 2 𝑉𝑖 1
= − + 𝐷𝑖
−
𝜕𝑡
𝜏
𝜕𝑋 2 𝐶
𝐿𝑘,𝑖
𝛾𝑖 2 𝜔 =
𝑁
𝐼𝑘,𝑖 − 𝐼𝑖𝑛𝑗 ,𝑖 ,
𝑘=1
𝑑𝐼𝑘,𝑖
= −𝑟𝑘,𝑖 𝐼𝑘,𝑖 + 𝑉𝑖 ,
𝑑𝑡
1 1
1
+𝜔+
𝐷𝑖 𝜏𝑖
𝐶𝑖
𝑘
𝑟𝑘,𝑖
1
.
+ 𝜔𝐿𝑘,𝑖
The nodes at which branches meet satisfy the continuity of potentials and the conservation of
current that passes through them. There are also two kinds of terminal node, open and closed
defined by the respective constraints that 𝑉𝑖 𝑙𝑖 , 𝑡 = 0 and 𝜕𝑉𝑖 𝑋, 𝑡 /𝜕𝑋 𝑋=𝑙 𝑖 = 0.
12
Similarly to the case of the infinite branch after rescaling the space by 𝑥𝑖 = 𝑋𝛾𝑖 (𝜔) we find the
frequency domain solution is derived to be
𝑙 𝑗 (𝜔)
𝑉𝑖 (𝑥, 𝜔) =
𝑗
0
𝑑𝑦𝐻𝑖𝑗 𝑥, 𝑦, 𝜔 𝐼𝑖𝑛𝑗 ,𝑗 (𝑦/𝛾𝑗 (𝜔), 𝜔)/𝐶𝑗 𝐷𝑗 𝛾𝑗 2 𝜔 .
All that remains is to determine the functions 𝐻𝑖𝑗 𝑥, 𝑦, 𝜔 which are constructed by a ‘sum-overtrips’ approach. Essentially 𝐻𝑖𝑗 𝑥, 𝑦, 𝜔 can be written in terms of 𝐻∞ :
𝐻𝑖𝑗 𝑥, 𝑦, 𝜔 =
𝐴𝑡𝑟𝑖𝑝 𝜔 𝐻∞ 𝑙𝑡𝑟𝑖𝑝 .
𝑡𝑟𝑖𝑝𝑠
𝑙𝑡𝑟𝑖𝑝 = 𝑙𝑡𝑟𝑖𝑝 (𝑥𝑖 , 𝑦𝑗 , 𝜔) is the length of a trip from point 𝑥 on branch 𝑖 that finishes at point 𝑦 on
branch 𝑗. For the intermediate branches 𝑘 distances are scaled appropriately by 𝛾𝑘 𝜔 . These trips
start at the point 𝑥𝑖 on branch 𝑖 and radiate away in either direction, being able to pass through
nodes or reflect back off them as well as terminals until they finally end up at the stimulus location
𝑦𝑗 . Trips are also allowed to pass through the input location but must eventually finish at it.
The coefficients of 𝐴𝑡𝑟𝑖𝑝 (𝜔) are chosen as follows:
1. From any starting point 𝐴𝑡𝑟𝑖𝑝 𝜔 = 1.
2. For every node through which the trip passes through from one branch 𝑘 onto a different
branch 𝑚 𝐴𝑡𝑟𝑖𝑝 (𝜔) is multiplied by a factor 2𝑝𝑚 𝜔 .
3. For every node off which the trip reflects back onto branch 𝑘, 𝐴𝑡𝑟𝑖𝑝 (𝜔) is multiplied by a
factor 2𝑝𝑘 𝜔 − 1.
4. For every closed (open) terminal node 𝐴𝑡𝑟𝑖𝑝 (𝜔) is multiplied by a factor +1 (-1).
The relevant branch parameters are defined as
𝑞𝑘 𝜔
,
𝑚 𝑞𝑚 𝜔
𝑝𝑘 𝜔 =
𝑞𝑖 𝜔 =
𝛾𝑖 𝜔
.
𝑅𝑎,𝑖
Checking Stability
We will now show that stability is satisfied given certain restraints on the branches in question, most
particularly that the membrane parameters are uniform throughout the branching structure, i.e.
𝑅, 𝐿, 𝐶 and 𝑟 are identical everywhere although the 𝐷𝑖 can be different for different branches 𝑖.
For a semi-infinite branch, the transfer function is
𝐻∞ 𝑋 − 𝑌 𝛾(𝜔)
𝐻∞ 𝑋 + 𝑌 𝛾(𝜔)
𝐻∞ 𝑋 − 𝑌 𝛾(𝜔) ± 𝐻∞ 𝑋 + 𝑌 𝛾(𝜔)
±
=
,
𝐷𝛾(𝜔)
𝐷𝛾(𝜔)
𝐷𝛾(𝜔)
where the ± is determined by whether the terminal is open or closed. However once again the
denominator is just 𝛾(𝜔) which once again only has zeros in the left half of the complex plane.
Therefore the transfer function is stable. For any arbitrary geometry where the membrane
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parameters are uniform for all branches then the coefficient terms 𝑝𝑘 𝜔 are all constant real
numbers. Therefore once again 𝛾(𝜔) is the only denominator term which could be zero for the
transfer function. In the case where the transfer functions are infinite sums simply note that the
exponential term in 𝐻∞ (𝑥) forces absolute convergence of the sums for all 𝜔 such that 𝛾(𝜔) ≠ 0.
Unfortunately the condition of uniform membrane parameters is typically unrealistic for many
neurons. Preliminary simulations indicate though that it might be the case that allowing different
membrane parameters on branches might not affect stability (Fig. 9B). Proving this analytically by
checking stability for the relevant transfer functions is a sensible extension for further research.
Another generalisation is the inclusion of more inductive branches as is possible in the membrane
circuit, which would correspond to adding more kinds of ion-channels to the membrane. For the
infinite branch we prove below that this should not affect stability and thus should allow for
immediate extension onto the case where membrane parameters are set to be uniform throughout
the neuron. The following also serves as a proof that the zeros of 𝛾 𝜔 only occur in the left half of
the complex plane.
Now by definition
𝐷𝛾 2 𝜔 = 𝜔 +
1
1
+
𝑅𝐶
𝐶
𝑁
𝑘=1
1
.
𝑟𝑘 + 𝐿𝑘 𝜔
𝐿𝑒𝑡 𝜔 = 𝑎 + 𝑏𝑖
2
⟹ 𝐷𝛾 𝑎 + 𝑏𝑖
=
1
1
+𝑎+
𝑅𝐶
𝐶
𝑁
𝑘=1
𝑟𝑘 + 𝐿𝑘 𝑎
𝑟𝑘 + 𝐿𝑘 𝑎 2 + 𝐿𝑘 𝑏
2
2
+ 𝐼𝑚(𝐷𝛾 2 𝜔 )
2
.
Now as 𝛾 𝜔 can only be zero when 𝛾 2 𝜔 is zero, then one can see from the above expression,
𝐷𝛾 2 𝑎 + 𝑏𝑖 can only be zero when 𝑎 < 0 as 𝑅, 𝐶, 𝐿𝑘 and 𝑟𝑘 are all positive. This implies that 𝛾 𝜔
can only be zero for 𝜔 in the left half of the complex plane. Hence the transfer function is stable.
Summary
Using the transfer function theorem an analytical solution for the system’s steady-state response to
a sinusoidal current has been derived and extended in part onto a more general branching structure.
The limitation of requiring uniform membrane parameters needs to be resolved for the method to
become realistically applicable to real neurons. The main challenge here is proving that the transfer
function is stable in general which may turn out to be a non-trivial complex analysis problem.
It appears that using the steady-state analysis to predict the response profile of the chirp poses two
difficulties. Firstly the chirp is never really in a steady-state, existing in a peculiar transient state.
Secondly the changing frequency of the chirp causes ‘knock-on’ effects (Fig. 9D) which may well lie
outside the scope of the frequency space analysis conducted here. Together these two points
suggest that using the chirp to determine the resonant frequency of a neuron may well fail to
accurately predict the frequency at which resonance occurs, as derived analytically.
As the transfer function for an arbitrary branching structure can be found using the trips approach
described here then other methods from circuit theory analysis should be available for studying this
14
system. Particularly it should be worthwhile using transient state analysis techniques to better
describe the initial response exhibited (Fig. 9A). This may also pave the way to a better
understanding of the profile generated by the chirp stimulus.
References & Bibliography
Coombes S, Timofeeva Y, Svensson CM, Lord GJ, Josic K, Cox SJ, Colbert CM (2007) Banching
dendrites with resonant membrane: a “sum-over-trips” approach. Biol Cybern 97:137-149
Abbott LF, Farhi E, Gutmann S (1991) The path integral for dendritic trees. Biol Cybern 66:49-60
Abbott LF (1992) Simple diagrammatic rules for solving dendritic cable problems. Phsysica A
185:343-356
Koch C (1984) Cable theory in neurons with active, linearized membranes. Biol Cybern 50:15-33
Koch C, Poggio T (1985) A simple algorithm for solving the cable equation in dendritic geometries of
arbitrary geometry. J Neuorosci Methods 12:303-315
Magee JC (1998) Dendritic hyperpolarization-activated currents modify the integrative properties of
hippocampal CA1 pyramidial neurons. J Neurosci 19:8219-8233
Mauro A, Conti F, Dodge F, Schor R (1970) Subthreshold behaviour and phenomenological
impedance of the squid giant axon. J Gen Physiol 55:497-523
Timofeeva Y, Lord GJ, Coombes S (2006) Dendritic cable with active spines: a modelling study in the
spike-diffuse spike framework. Neurocomputing 69:1058-1061
Ulrich D (2002) Dendritic Resonance in Rat Neocortical Pyramidal Cells. J Neurophysiol 87: 2753–
2759
Narayanan R and Johnston D (2008) The h Channel Mediates Location Dependence and Plasticity of
Intrinsic Phase Response in Rat Hippocampal Neurons. J Neuroscience 28(22):5846 –5860
DeCarlo RA, Lin Pen-Min (1995) Linear Circuit Analysis Time Domain, Phasor, and Laplace Transform
Approaches Second Edition. Oxford University Press
Carnevale NT, Hines ML (2006) The NEURON Book. Cambridge University Press, London.
Ernst Niebur (2008) Neuronal cable theory. Scholarpedia 3(5):2674
http://www.scholarpedia.org/article/Cable_theory#Telegrapher1
Dr. Clay M. Armstrong (2008) Gating currents. Scholarpedia
http://www.scholarpedia.org/article/Gating_currents
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