# A two-variable model of somatic-dendritic interactions in a bursting neuron.

**ABSTRACT** We present a two-variable delay-differential-equation model of a pyramidal cell from the electrosensory lateral line lobe of a weakly electric fish that is capable of burst discharge. It is a simplification of a six-dimensional ordinary differential equation model for such a cell whose bifurcation structure has been analyzed (Doiron et al., J. Comput. Neurosci., 12, 2002). We have modeled the effects of back-propagating action potentials by a delay, and use an integrate-and-fire mechanism for action potential generation. The simplicity of the model presented here allows one to explicitly derive a two-dimensional map for successive interspike intervals, and to analytically investigate the effects of time-dependent forcing on such a model neuron. Some of the effects discussed include 'burst excitability', the creation of resonance tongues under periodic forcing, and stochastic resonance. We also investigate the effects of changing the parameters of the model.

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A two–variable model of somatic–dendritic interactions in

a bursting neuron

Carlo R. Laing and Andr´ e Longtin

Department of Physics, University of Ottawa, 150 Louis Pasteur, Ottawa, ON,

Canada K1N 6N5.

Submitted to Bulletin of Mathematical Biology

November 16, 2001.

Resubmitted March 4, 2002.

Running title: Model of a bursting neuron.

Corresponding author:

Carlo R. Laing

Department of Physics,

University of Ottawa,

150 Louis Pasteur,

Ottawa, ON,

Canada K1N 6N5.

ph: (613) 562 5800 extn 6744

fax: (613) 562 5190

email: claing@science.uottawa.ca

We would like to submit the final manuscript electronically in LaTeX format.

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A two–variable model of somatic–dendritic interactions in a bursting

neuron

Abstract

We present a two–variable delay–differential–equationmodel of a pyramidal cell from

the electrosensory lateral line lobe of a weakly electric fish that is capable of burst dis-

charge. It is a simplification of a 6-dimensional ordinary differential equation model for

such a cell whose bifurcation structure has been analyzed (Doiron et al., J. Comput. Neu-

rosci., 12(1), 2002). We have modeled the effects of back–propagating action potentials

by a delay, and use an integrate–and–fire mechanism for action potential generation. The

simplicity of the model presented here allows one to explicitly derive a two–dimensional

map for successive interspike intervals, and to analytically investigate the effects of time–

dependent forcing on such a model neuron. Some of the effects discussed include “burst

excitability”, the creation of resonance tongues under periodic forcing, and stochastic res-

onance. We also investigate the effects of changing the parameters of the model.

1Introduction

Bursting, in which a cell periodically switches from quiescent behavior to a rapidly spiking

state and back again, is an important and common form of behavior [6, 19, 22, 28, 29, 36]. A

number of mathematical models of bursting cells have been developed [1, 6, 7, 36, 39, 47], but

they are generally difficult to treat analytically in any detail. Much past analysis of bursting

cells has been influenced by the “slow–fast” separation of time–scales in bursting systems[19,

36], where it is assumed that fast, spiking variables act on a much shorter time–scale than the

slow variable(s) that are responsible for the shifts between spiking and quiescent behavior.

Recently, a new mechanism for burst discharge in pyramidal cells of the weakly electric

fishApteronotus leptorhynchus wasinvestigated[7]. Thesecells receiveinputdirectlyfromelec-

troreceptor cells on the fish’s skin, and are thought to play a significant role in the processing

of electrosensory information. The model presented in [7] was a set of six coupled nonlinear

first–order ordinary differential equations, which was a reduction from the multicompart-

ment model involving over 1500 variables presented in [8]. This reduction was obtained by

lumping the many compartments into two, representing the soma and the dendrite, and by

ignoring the dynamics of the channels not thought to be important in the mechanism for

bursting. That the model in [7] reproduced both the bursting behavior observed in the model

of [8] and experimentally observed bursts [28] indicates that this process was successful.

The model analyzed in [7] was studied using the “slow–fast” approach of others [19, 36],

but it differed from all previous bursting models in that when the one slow variable was

held constant, the remaining “fast” system did not show bistability for any values of the slow

variable. The bifurcation in the fast system that ended a burst was found to be a transition

from period–one to period–two behavior associated with the failure of a somatic action po-

tential to induce a dendritic one, and the interburst interval was found to involve the passage

in phase space near a fixed point. Several aspects of the timing of bursts were found to be

related to the distance in parameter space from a saddle–node bifurcation, hence the name

“ghostbursting” [42].

In this paper we further reduce the model in [7] to a set of two discontinuous delay dif-

ferential equations, from which a two–dimensional map can be derived (assuming constant

input current). This gives us analytical insight into complex soma–dendrite interactions, and

is computationally much easier to study than an ODE model. The reduced model resented

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here can be constructed because, as a result of the work in [7], we understand the essential

ingredients of this type of bursting (refer to Figure 1). A short time after most somatic spikes,

current flows from the dendrite to the soma, producing a depolarizing afterpotential (DAP)

at the soma. For large enough current injected to the soma, the sizes of these DAPs slowly

increase due to a slow inactivation of the dendritic potassium that is responsible for the re-

polarization of dendritic action potentials. This results in progressively smaller inter–spike

intervals (ISIs), and this process continues until an ISI is smaller than the refractory period

of the dendrite. Once this happens, there is dendritic spike failure, which removes the nor-

mal current flow to the soma, and a DAP does not appear. This results in a long ISI, during

which the variable controlling inactivation ofdendriticpotassiumincreases, and the sequence

starts again. In Figure 1 we show typical bursting behavior from the model in [7]. The spike

patterning is similar to that seen in the multicompartment model in [8] and in experimental

recordings [28].

If we considergradually increasing the DC current injected into the soma ofsuch an actual

pyramidal cell, it is observed that the cell’s behavior changes from quiescent to periodic firing

and then to bursting [9, 28]. This behavior is also seen in the pyramidal cell models [7, 8] and

the model presented below, and is in marked contrast with many other burst mechanisms for

which a cell follows the pattern quiescence, bursting, tonic, as the current is increased [33, 36,

41, 43].

In this paper we also consider periodically modulating the current applied to the model

neuron, and forthecase ofsinusoidalmodulationweobtain athree–dimensionalmap forsuc-

cessive spike times. This map can be used to determine the boundaries in parameter space of

resonance tongues,in which the neuron’s firing frequency is locked to that of the forcing. This

map is able to explain some of the behavior seen in [26], in which bursting models (including

the one in [7]) are periodically forced. For example, when only a DC current is injected into

the model neuron of [7], there is a value of current at which the neuron switches from tonic to

bursting behavior. However, adding a sinusoidal modulation to the injected current can ei-

ther increase or decrease the DC value of the current where the transition occurs. The amount

of increase or decrease depends on the frequency of the modulation. The simplified model

presented here can also reproduce the phenomenon of “burst excitability” that is explored in

more detail in [25], and by varying parameters, the “gallery” of burst types seen in [7].

The two–dimensional model that we have developed would be very useful for large–

scale simulations of networks of such neurons as it has fewer variables than common neuron

models that involve ionic channels [7], the differential equations involved are not stiff, and

its piecewise linear nature aids its analysis. The way we have modeled backpropagation of

an action potential along a dendrite and the resulting “ping–pong” effect [48] by a discrete

delay in a low–dimensional system is also novel. (The ping–pong effect refers to the interplay

between somatic and dendritic action potentials and the electrotonic currents that flow as a

result of them not occurring at the same time, nor being of the same duration.) There has

been a great deal of recent interest in backpropagation in active dendrites [18, 38, 46], in

relation to the processing of synaptic inputs and the induction of synaptic plasticity [20].

The modeling of active dendrites presented here could possibly be applied to these systems

in which backpropagation in active dendrites (or the presence of a second compartment in a

two–compartment model [2, 23, 30, 33]) is important, provided that the effects of the dendrite

on the firing pattern of the soma are understood.

In Section 2 we present the model, derive the corresponding map and investigate its prop-

erties, including the effects of changing its parameters. In Section 3 we present the sinu-

soidally forced model, derive the corresponding map, and discuss resonance tongues and

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stochastic resonance. Section 4 is a summary and discussion. The appendix contains a dis-

cussion of other possible models that show “ghostbursting”. The point of this is to show that

there is not one unique model of this type of behavior, but rather a variety, each of which has

its own advantages and disadvantages.

2The Model

We use an integrate–and–fire neuron [21] to produce somatic action potentials (“spikes”),

and couple this with another variable, c, whose behavior mimics the effects of inactivation of

dendritic potassium in the model of [7] (although c increases during a burst, while the actual

inactivation gating variable decreases in [7] (see Figure 1); both trends have the same effect

in their respective models). The effective delay between a somatic spike and the appearance

of a DAP (a result of the diffusive coupling in voltage between soma and dendrite in [7])

is mimicked by an actual delay, and the effect of the DAP is mimicked by an instantaneous

increase in the neuron’s voltage.

The equations are

soma, H( ?) is the Heaviside function, r represents the refractory period of the dendrite,

the effective delay between the somatic action potential and the dendritic–to–somatic current

that causes the DAP, and A

occurring at the times tn.

At almost all times, V exponentially approaches I from below with time–constant 1, and c

exponentially decays towards 0 with time–constant

c

greater than the refractory period r, V is incremented: V

time

Note that the neuron will not fire if I is always less than 1.

An example of the behavior of (1)-(2) is shown in Figure 2. Note the increase in the overall

level of c and the decrease of ISIs during the burst, and the long ISI separating bursts. This

long ISI is a result of the previous ISI being less than the refractory period of the dendrite of

the neuron, so that no current propagates from the dendrite to the soma during this ISI. This

long ISI can also be seen as the smallest instantaneous frequency in Figure 3 for I greater than

dV

dt

dc

dt

?

I

? V

? Ac∑

n

H(tn

?tn

?1

?r) Æ(t

? tn

??)(1)

??c

???(B

?Cc2)∑

n

Æ(t

?tn)(2)

with the rule V(t

is an integer). V represents the somatic membrane potential, I is the current injected to the

?

n)

? 0 if V(t

?

n)

? 1, and the tnare the times at which the reset occurs (n

? is

? B

?C and

? are constants. The action potentials are thought of as

?. At each firing time tn, c is incremented:

?? c

? B

? Cc2. At a time

? after firing, and assuming that the previous ISI, tn

? tn

?1, is

?? V

? Ac, where c is evaluated at a

? after firing. If the previous ISI is less than the refractory period, V is not incremented.

? 1 ?22. This bursting behavior is quite robust with respect to changes in parameters.

The model (1)-(2) has not been formally derived from any other model, but has been con-

structed as a result of understanding the ionic mechanisms behind ghostbursting [7, 8]. As

mentionedin the Appendix,a number ofequally valid models could be used. It may be possi-

ble to construct similar models for other types of neurons, provided the interactions between

the soma and dendrites are understood.

We now derive an exact map describing the behavior of (1)-(2).

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2.1Derivation of the map

Because of the linearity of (1)-(2) for values of t at which the

to derive a map for interspikeintervals and values of c just after firing. Recall that the solution

of (1) when A

V(t)

Æ functions are zero, it is possible

? 0 is

? I

?[V(0)

? I]e

?t

(3)

Assume that t

? t

?

n, V(t

?

n)

? 0, c(t

?

n)

? cn, and tn

n, when V

?tn

?1

? r. (V(t

?

n) is defined as lim

??0V(tn

?

?), where

by Acne

V continues to evolve, reaching 1 after a further time s, where

?? 0.) At a time

? after t

?

? I(1

? e

??) and c

? cne

????, V is incremented

????. Assuming that this does not push V above 1, i.e. that I(1

?e

??)

? Acne

????

? 1,

I

?[I(1

?e

??)

? Acne

????

? I]e

?s

? 1(4)

(s

? 0 when t

? tn

??). Solving (4) for s we obtain

s

? ln

?

Acne

????

? Ie

??

1

? I

?

(5)

Alternatively, if tn

so there is no feedback from dendrite to soma), V will reach 1 after a time

?tn

?1

? r (i.e.thelast ISIwas lessthantherefractoryperiodofthedendrite,

ln

?

I

I

?1

?

?

(6)

measured from tn. During the interval (tn

and is then updated at time tn

a piecewise two–dimensional map for ∆n

?tn

?1), c exponentially decays with time constant

?,

?1. Thus, taking all possible cases into consideration, we have

?1

? tn

?1

?tnand cn

?1in terms of ∆nand cn:

∆n

The instantaneous frequency, i.e. 1 ?∆n, is shown as a function of I in Figure 3 for a partic-

ular set of parameter values (transients have been removed). A number of observations can

be made:

?1

?

?

?

?

?

?

?

if ∆n

if ∆n

if ∆n

? r and I(1

?e

? ?)

? Acne

??? ?

? 1 (i)

??ln

?

Acne

?? ??

?Ie

??

1 ?I

?

? r and I(1

?e

??)

? Acne

????

? 1 (ii)

ln

?

I

I

?1

?

? r

(iii)

(7)

cn

?1

?

cne

?∆n ?1

??

? B

?C[cne

?∆n ?1

??]2

(8)

This is all under the assumption that I(1

this inequality is not satisfied, e.g. if I is too large, then ∆n

above. As can be seen from (8), the small value of ∆n

of cito increase without bound, an unphysical situation. Note that we must have

? e

??)

? 1, i.e. that neither I nor

? are too large. If

?1

? ln[I

?(I

?1)] as in case (iii)

?1in this situation may cause the values

?? r.

1. During periodic firing for 1

firing, cn

of thesteady–stateperiod, ∆ (the negative square root must be chosen). Substitutingthis

into case (ii) in (7), we obtain an equation that ∆ must satisfy:

? I

?? 1 ?22, case (ii) in (7) will always be true. For periodic

?1

?cn, and thus(8) is aquadraticin cn. Itcanthereforebe solvedforcninterms

(1

? I)e

??e∆

?

Ae

????

?

1

?e

?∆

??

?

?

1

?2e

?∆

??

?(1

?4BC)e

?2∆

??

?

2Ce

?2∆

??

? Ie

??

(9)

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For 1

can be seen in Figure 3; the unstable solution is not shown). The two solutions coalesce

in a saddle–node bifurcation [24] at I

bifurcation of periodic orbits. Interestingly, it was a saddle–node bifurcation of periodic

orbits that separated periodic from bursting behavior in the full ionic ODE model in [7].

As I

tending to zero. Note that if

? I

?? 1 ?22, equation (9) has two solutions, the larger one of which is stable (and

? 1 ?22. Note that in (1)-(2), this is a saddle–node

? 1 from above, the largest root of (9) tends to

?, corresponding to the frequency

?? 1, (9) is independent of

?.

2. For I greater than

where the only current driving the neuron during its entire period is I. Thus the lower

curve in Figure 3 for I greater than

? 1 ?22, the smallest instantaneous frequency occurs between bursts,

? 1 ?22 is just ∆

? ln[I

?(I

?1)] (case (iii) in (7)).

3. For I greater than

burst interval is bounded below by the curve

? 1 ?22, the “band” of frequencies in Figure 3 not including the inter-

∆

???ln

?

ABe

????

? Ie

??

1

? I

?

(10)

since cnhas a minimum value of B after being reset at the end of the inter–burst interval.

(Expression (10) is obtained by setting cn

? B in (7) (ii)).

4. For I greater than

the longest ∆nis a function of I. Note that for I greater than

nates between a long ISI and a short ISI, which can be thought of as periodically firing

“doublets”.

? 1 ?45, the map alternates between cases (i) and (iii) in (7), so only

? 1 ?41 the system alter-

5. The progression quiescence

ISIs as I is increased as seen in Figure 3 is also seen in the models of [7, 8], and experi-

mental recordings [28, 45].

? periodic firing

? bursting

? alternating short and long

In relation to point 1 above, if the positive square root is taken instead and substituted into

case (ii) in (7), the rootsof the resulting function may not satisfy the condition ∆

with case (ii), and thus they will not be actual fixed points of the map. Also, the fact that (8)

can be solved explicitly for the steady state value of c is a result of our choice of the dynamics

of c, (2). Replacing the term Cc2in (2) by Cc would mean that the equation for the steady

state of c was linear and thus had only one solution; this choice would also simplify the

expression (9), but would make chaotic behavior more difficult, although not impossible, to

obtain (see below).

? r associated

2.2Lyapunov exponent

The Lyapunov exponents of a trajectory determine its stability and the behavior of nearby

trajectories. If a stable solution has at least one positive Lyapunov exponent, the system will

exhibit sensitivity to initial conditions, and nearby trajectories will typically separate expo-

nentially in time [10]. For a range of current values, the bursting behavior in [7] was shown

to have a positive Lyapunov exponent, and thus be chaotic.

To find the maximal Lyapunov exponent,

cally calculate the Jacobian, Df, of (7)-(8) and evaluate it at each point on the orbit, x1

?, for a trajectory of the map (7)-(8), we analyti-

?x2

????,

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where xi

calculated [10] from

? (∆i

?ci)

? R2. If qiis the largest magnitude eigenvalue of Df(xi) then

? can be

?? lim

n

??

1

n

n

∑

i ?1

ln

?qi

?

(11)

This quantity is shown in Figure 4, multiplied by 4 for clarity, together with instantaneous

frequency. Note the transition from period–1 firing to chaotic bursting at I

long–period quasiperiodic behavior for I greater than

? 1 ?22, and the

? 1 ?32.

2.3Effects of parameters

The model (1)-(2) has six parameters that are regarded as constant (A

now briefly discuss the effects of changing each of these. In more realistic models [7, 8],

changing parameters can mimic the application of drugs that selectively block various ionic

channels (e.g.tetrodotoxinis a Na

nel properties by the fish itself. While the relationships betweenthe six parameters below and

the many parameters of more realistic neurons are not yet known, knowing how changes in

them affect the model’s behavior is still of interest (see below).

?r

????? B and C). We

?channel blocker[28]), orthe dynamic modulationofchan-

? A mimics the size of the current flowing from dendrite to soma, and thus the size of the

DAP and how much it contributes towards moving the soma to spike threshold. Thus

increasing A will make the neuron more likely to burst.

? r is irrelevant during periodic firing (note that (9) is independent of r), so changing it

cannot switch the neuron from periodic to bursting or vice versa. However, decreasing

r makes the bursting more likely to be chaotic. This is because c can now reach a larger

value at the end of a burst before it is terminated, and it is the nonlinear growth of c at

the end of the burst that is the source of the chaos. However, if c can take on large values

at the end of a burst, there may not be time during the interburst interval for c to decay

sufficiently before the beginning of the next burst, and this can lead to an unphysical

“blow–up” in c. Note that r must be greater than

?.

?? governsthe rate ofdecay of c. Whethertheneuronfires periodically orburstsdepends

on the balance between the decay of c between spikes and the increment in c at each

spike. As seen from (8), increasing

more likely to burst. Note from (1)-(2) that the time–constants governing the dynamics

of V and c are 1 and

? causes c to decay more slowly, making the neuron

? respectively, and that it is possible for bursting to occur with

?? 1 (not shown). This is in contrast to the usual analysis of bursting, where there is

assumed to be a separation betweenthe fast spiking dynamics and the slower dynamics

that control the burst length and interburst intervals [19, 36].

Also, as in [7], there is a range of

is too small, c decays too much between action potentials, and the slow growth during

a burst does not occur. Similarly, if

successive maxima of c grow extremely quickly. This leads to “doublet” firing in V and

an eventual breakdown of the algorithm.

? values for which bursting is seen as I is varied. If

?

? is too large, c cannot decay between bursts and the

? For the parameters used, if

and vice versa. If

versa. This can be understood graphically by writing (9) as

?? 1, increasing

? makes the neuron more likely to burst

?? 1, increasing

? makes the neuron less likely to burst, and vice

e

??f(∆)

? Ae

????g(∆

??)(12)

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where

f(∆)

? I

?(1

? I)e∆

and

g(∆

??)

?1

? e

?∆

??

?

?

1

?2e

2Ce

?∆

??

?(1

?4BC)e

?2∆

??

?2∆

??

(13)

For a given

function of ∆, and g(∆

region of interest. The intersections of the left and right sides of (12) give the values

of ∆ at the two periodic orbits (one stable and the other unstable) of (1)-(2). It is the

coalescence ofthesein asaddle–nodebifurcation that marksthetransitionfromperiodic

to bursting behavior.

? and other parameters in the appropriate range, f(∆) is a concave–down

??) is a concave–up function of ∆. They are both positive in the

If

right hand side, bringing the two points of intersection closer to one another, and thus

lowering the value of I at which the saddle–node bifurcation occurs. Conversely, if

?? 1, increasing

? decreases the left hand side of (12) more than it decreases the

?? 1, increasing

hand side, moving the two points of intersection further apart and thus increasing the

value of I at which the saddle–node bifurcation occurs.

? decreases the right hand side of (12) more than it decreases the left

? B and C both control the increment in c at each spike, so increasing either will make the

neuron more likely to burst. The Cc2term is not strictly necessary to obtain bursts, as

we can produce a plot like Figure 3 with C

not chaotic). However, without this nonlinear term it is difficult to obtain chaotic bursts.

This is because it is the nonlinear growth of c at the end of a burst, and the fact that the

largest value of c during a burst is carried over to the start of the next burst, that cause

chaotic dynamics to occur. If B

having a nonzero value of B prevents this.

? 0 (although the bursts are then periodic,

? 0 then cn

? 0 for all n is a possible solution of (8), and

This knowledge is useful, since if we can determine the qualitative relationship between pa-

rameters in a more realistic model of this cell (e.g. the models in [7, 8]) and the six parameters

above, we can understand how changing the parameters in the more realistic models will

change the behavior of the cell, without having to simulate those more detailed models. To

determine these relationships, one would need to know the effects of changing a particular

parameter in a large model on one or more of the six parameters above.

As an example, it was found in [7] that decreasing the maximum conductance of the den-

dritic potassium decreasedthe value of current injected to the soma at which the cell switched

from tonic to bursting, i.e. it made the cell more likely to burst. This is easy to understand,

since it is dendritic potassium that is responsible for repolarizing the dendrite, and by lessen-

ing its effect the dendritic action potential is widened, leading to a larger DAP at the soma.

Thus, decreasing this conductance is equivalent to increasing A in (1)-(2). By the same reason-

ing, decreasing the maximum conductance of the somatic potassium in the model of Ref. [7]

is equivalent to decreasing A, making the cell less likely to burst. A similar result regarding

the effects of changing the somatic–to–total area ratio [27] can be explained in a similar way.

One of the parameters in actual pyramidal cells that is thought to change over time is the

contribution of slow voltage–activated persistent sodium in the dendrite [9]. This becomes

relevant when a cell is depolarized for a long time (of the order of 1 second). Another pa-

rameter that is thought to change on a relatively fast time–scale (although still much more

slowly than the time–scale of action potential production) is the maximum conductance of

dendritic potassium, whose kinetics are subject to second messenger regulation [45]. Thus,

making the link between parameters in real or realistic model neurons and the six parameters

of the model presented here will increase the usefulness of this model.

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2.4 Burst excitability

“Burst excitability” [25] is a type of excitability analogous to “normal” excitability [13, 17, 19,

36, 38], except that the “event” that follows a sufficiently strong transient excitation is a burst

ratherthan a single action potential, and the systemmay returnto periodicfiring after a burst,

rather than quiescence.

The presence of a saddle–node bifurcation of fixed points of the map (7)-(8) marking

the transition between periodic and burst behavior implies burst excitability at this bound-

ary [25]. For a value of I less than that at the bifurcation, a temporary step increase in I may

induce a burst in which the firing rate is elevated from its prior value for a time greater than

the duration of the increase in I. If the magnitude of the increase in I is not sufficient, or its

duration is too short, a burst will not be observed. Such behavior is shown in Figure 5. Note

that the length of the burst in Figure 5, top, (i.e. the time between the perturbation of I and

the doublet marking the end of the burst) is about four times the length of the longest time

constant in the system ( ?

even longer. (There is no bistability in the system, and the long period of higher firing fre-

quency is a result of the perturbation pushing the system very close to an unstable periodic

orbit.) This reinforces the idea presented in [7] that “slow” behavior in a bursting system

does not necessarily imply the existence of a slow time–scale in the form of an explicit long

time–constant, but can be the result of the system’s trajectory in phase space passing close to

the stable manifold of an unstable object (e.g. a fixed point or periodic orbit).

This form of burst excitability has been seen in the model presented in Ref. [7]. Since the

pyramidal cells we are modeling receive sensory input directly from electroreceptors on the

fish’s skin [3, 31], burst excitability may be a robust way of signaling a transient increase in

the strength of the electric field at those electroreceptors caused by, for example, a “chirp”

emitted by another fish [50].

? 1). The recovery from the perturbation in Figure 5, bottom, is

2.5A “gallery” of bursts

We know for the system (1)-(2) that the current threshold for spiking is I

length of an interburst interval is given by ∆

tends to 1 from above. Also, since the bifurcation separating periodic from bursting behavior

is a saddle-node bifurcation of periodic orbits, standard results regarding type–I intermit-

tency [34] show that the number of spikes in a burst scales as

? 1, and that the

? ln[I

?(I

?1)], which tends to infinity as I

N

?

1

?I

? Isn

(14)

as I tends to Isnfrom above, where Isnis the value of I at which the saddle–node bifurcation

occurs ( ? 1 ?22 in Figure 3). (Type–I intermittency may occur when there is a saddle–node

bifurcation of periodic orbits in a system, i.e. a stable and unstable periodic orbit collide as

a parameter is varied. Just after the two orbits have collided there is a “trapping region” in

phase space where the system’s behavior is almost periodic. The amount of time spent in this

region scales as the reciprocal of the square–root of the distance in parameter space from the

bifurcation, hence the expression (14).)

The value of Isndepends on the other parameters in the system, but if we can change

them so that Isnmoves relative to the spiking threshold of 1 (and in particular, if we can

make it approach 1), then we will be able to obtain a wide variety of different burst lengths

and interburst intervals, as was done in [7]. In Figure 6 we show such a situation, where we

9

Page 10

have chosen to vary the parameter B. We can see that the curve separating periodic from

burst firing touches the curve separating quiescence from periodic firing (the line I

B

potassium was varied. Note that the curve in this Figure can be found numerically from (9)

by determining the value of I as a function of B at which the two roots of (9) coalesce; the

map, (7)-(8), does not have to be iterated.

By choosing different points in Figure 6 we can obtain different bursts — this is shown in

Figure 7. In the top panel, we are close to the curve of saddle–node bifurcations of periodic

orbits, but far from I

interburstintervalis notparticularly so. Inthe middlepanel, weare close tothecodimension–

two point in Figure 6 (marked with a circle), so both the bursts and the interburst intervals are

long. In the bottom panel we are to the right of the codimension–two point in Figure 6, and

(as was observed in [7]) we observe doublets, i.e. a small ISI followed by a larger one. This is

easily understood, since the first spike induces a large DAP (large because B is large) which

induces a second spike. This second one marks the end of an ISI that is within the refractory

period of the dendrite, so no DAP arrives after it and a long ISI is produced (long because I

is close to 1).

It is possible that the bursts produced by these cells are of most interest, rather than the

individual spikes within them [29]. Thus being able to change both the number of spikes in a

burst and the length of the interburst interval, as we have just done, may be very important

with respect to changing the information content of the output of such a cell [7].

One difference between the model (1)-(2) and the ODE model of [7] involves the scaling

of the interburst intervals as I decreases. The bifurcation separating quiescence from periodic

firing in [7] is a saddle–node–on–a–circle bifurcation [24], and hence the period of periodic

firing scales as T

takes place. This is in contrast with the ∆

mechanism that we are using in (1)-(2) to produce action potentials. Thus in the interburst

intervals, where there is essentially no current flowing from the dendrite to the soma and the

soma is drivenby only the current injected into it from the outside, the lengthofthe interburst

intervals will scale differently with current for the two models. However, both scalings give

the same qualitative result, i.e. T

The scaling in (1)-(2) could be made to match the scaling in [7] if the spike–producing neuron

was one whose bifurcation separating quiescence from periodic firing was a saddle–node–

on–a–circle bifurcation; one example is the “theta neuron” [17], but its nonlinearity would

complicate analysis of the resulting bursting model.

? 1) near

? 0 ?42. This is similar to the situation in [7], where the maximum conductance of dendritic

? 1. Hence the bursts are long (compare with those in Figure 2) but the

? 1 ?

?I

? I

?, where I

?is the value of the current at which the transition

? ln[I

?(I

?1)] expression for the integrate–and–fire

?? as I

? I

?from above, and ∆

?? as I

? 1 from above.

3 Sinusoidal forcing

The responseofdynamical systemstoperiodicforcing is a widely studiedproblem. Examples

with biological motivation include [4, 5, 13, 14, 16, 21, 40]. We will now consider the situation

where I in (1) is sinusoidally modulated. The modulation has amplitude Γ and angular fre-

quency

tric fish, e.g. Apteronotus leptorhynchus, continuously generate an approximately sinusoidal

modulation of the electric field surrounding them via their electric organ discharge (EOD).

The frequency of this oscillation can be as high as 1200 Hz and is essentially constant for any

particular fish. Males have higher frequencies than females. When two fish are physically

close, the difference in EOD frequencies gives rise to a “beat” frequency, equal to the differ-

?. This form of forcing is relevant for the following reasons. Wave–type weakly elec-

10

Page 11

ence between the two EOD frequencies. Thus there are at least two types of sinusoidal inputs

(the fish’s own EOD and the beating oscillation) that may be detected by electroreceptors on

the fish’s skin [3, 31] and passed to the pyramidal cells that we are modeling.

We will now derive a map similar to (7)-(8) for the case where I in (1) is sinusoidally mod-

ulated. Since the dynamics are no longer invariant with respect to time translation, we find

that they can only be reduced to a three–dimensional map, rather than a two–dimensional

one as above.

Assume that t

n, V(t

? t

??

n)

? 0, c(t

?

n)

? cn, and that tn

[sin( ?tn)

?tn

?1

? r. V evolves under

dV

dt

? I

? V

?Γsin( ?t)(15)

which has the solution

V(t;tn)

?

I[1

? e

?(t ?tn)]

[sin( ?tn)

?

?

Γ

1

??2

?

?sin( ?t)

??cos( ?t) ?

?

?Γe

?(t ?tn)

1

??2

?

??cos( ?tn)](16)

Note that this satisfies V(tn;tn)

?

? 0. At a time

? after tn,

V

? V

?

I(1

?e

??)

[sin( ?t)

?

?

Γ

1

??2

?

?sin[ ?(tn

??)]

??cos[ ?(tn

??)] ?

?

?Γe

1

??

??2

?

??cos( ?tn)](17)

and c

an amount Acne

for firing, then tn

V

? cne

????. (We assume that V(t;tn)

[sin( ?t)

? 1 for tn

? t

? tn

??.) V is now incremented by

??? ?. If V

? Acne

????

? 1, i.e. if this increment pushes V over the threshold

?1

? tn

??. If this is not the case, we need to solve (15) with V(tn

??)

?

? Acne

???. This has the solution

V(t;tn

?cn)

? I

?

?

Γ

1

??2

?

??cos( ?t)]

?e

?(t ?tn

??)

?

V

? Acne

????

? I

?

?

Γ

1

??2

?

?sin[ ?(tn

??)]

??cos[ ?(tn

??)] ?

?

(18)

?

I

?

?

Γ

1

??2

?

??cos( ?t)]

?e

?(t ?tn

??)

?

Acne

????

? Ie

??

?

?Γe

1

??

??2

?

[sin( ?tn)

[sin( ?tn

??cos( ?tn)]

?

(19)

and tn

?1is the smallest solution greater than tnof

V(tn

[sin( ?tn)

?1;tn

?cn)

? 1(20)

All of the above is under the assumption that tn

current flowing from the dendrite to the soma, then from (16), tn

(greater than tn) of

? tn

?1

? r. If tn

? tn

?1

? r, i.e. there is no

?1is the smallest solution

1

?

I

?

1

?e

?(tn ?1

?tn)

?

?

?

Γ

1

??2

?

?1)

??cos( ?tn

?1)]

?

?Γe

?(tn ?1

?tn)

1

??2

?

??cos( ?tn)](21)

11

Page 12

During the time interval (tn

a piecewise map for tn

?tn

?1), c decaysexponentially withtime–constant

?. Thus we have

?1and cn

?1in terms of tn

?tn

?1and cn:

tn

?1

?

?

?

?

tn

??

if tn

if tn

if tn

?tn

?1

? r and V

? Acne

????

? 1

min?s

? tn: V(s;tn

?cn)

? 1 ??tn

?1

? r and V

? Acne

????

? 1

min?s

? tn: Λ(s;tn)

? 1 ??tn

?1

? r

(22)

cn

?1

?

cne

?(tn ?1

?tn) ??

? B

?C

?

cne

?(tn ?1

?tn) ? ?

?2

(23)

where Λ(s;tn)

We can write the map (22)-(23) as

? V(s;tn) and V(s;tn) is given by (16), and V(s;tn

?cn) is given by (19).

tn

cn

?1

?

f(tn

g(tn

?tn

?1

?cn)(24)

?1

?

?1

?tn

?cn)

? g(f(tn

?tn

?1

?cn) ?tn

?cn)

? h(tn

?tn

?1

?cn)(25)

or

tn

sn

cn

?1

?

f(tn

tn

h(tn

?sn

?cn)(26)

?1

?

(27)

?1

??sn

?cn)(28)

i.e. a map from R3to R3, where

g(a

?b

?c)

? ce

?(a ?b) ??

? B

?C

?

ce

?(a ?b) ??

?2

(29)

and f is given by (22).

Note that when Γ

? 0, min

?s

? tn: V(s;tn

?cn)

? 1 ? has the solution

s

? tn

???ln

?

Acne

????

? Ie

??

1

? I

?

(30)

and min

reduces to (7)-(8), which only involves time differences, as is expected from a time–trans-

lationally invariant system. The nonzero amplitude of forcing in (15) breaks this invariance.

Notealsothat themap (26)-(28) hasnofixedpoints,asthevariables sand tdenotefiringtimes,

not interspike intervals. The maximal Lyapunov exponent for (26)-(28) can be calculated in a

similar way to that for the two–dimensional map (7)-(8) (see Section 2.2).

?s

? tn: Λ(s;tn)

? 1 ? has the solution s

? tn

? ln[I

?(I

?1)], and the map (22)-(23)

3.1Arnold tongues

Much work has been done on periodically forced oscillators [4, 13, 14, 16], and it is well–

known that an oscillator can become entrained to the frequency of the forcing. This may be of

relevance for the model we are considering, since (as mentioned above) the pyramidal cells

that we are modeling receive direct afferent input from electroreceptors on the fish’s skin.

This input has periodic components, and if a pyramidal cell was entrained to its inputs, it

could faithfully track that frequency over some range. Periodically moving between the tonic

state and a bursting state (so that, e.g. a burst always terminated at a fixed phase of the input)

may also be a mechanism for robustly signaling a periodically changing input.

12

Page 13

Regions of parameter space in which the frequency of the forcing and the frequency of

the forced oscillator have a particular ratio are called “Arnold tongues” [14], and are labeled

by the ratio of frequencies, e.g. 3 : 2. (For this example, the oscillator would pass through 2

cycles in the same time that the forcing signal took to pass through 3 cycles.) The system (1)-

(2) is capable of periodically oscillating for some values of its parameters and input, so under

periodic forcing we expect it to have some features in common with periodically forced oscil-

lators. However, the presence of the bifurcation separating periodic from burst firing in the

unforced system may mean that new features appear when it is periodically forced.

[sin( ?tn

Considerthecase of q : 1 locking, where we have 1 firing during a periodof qT ( ?

i.e. during q forcing cycles. For this case, cn

? 2 ??T),

?1

? cn, so let c

?be the smallest root of

[sin( ?tn)

cn

? cne

?qT

??

? B

? C

?

cne

?qT

??

?2

(31)

We can see from (19) that in this periodically–locked state

sin( ?tn

1

?

I

?

?

Γ

1

??2

?

?1)

??cos( ?tn

?1)]

?

e

?(tn ?1

?tn

??)

?

Ac

?e

????

? Ie

??

?

?Γe

1

??

??2

?

??cos( ?tn)]

?

(32)

Noting that for this locked state

?1)

??cos( ?tn

?1)

? sin( ?tn)

??cos( ?tn)(33)

and defining the firing phase in a similar way to Coombes and Bressloff [4] as

?n

??

?

tn

? Tint

?tn

T

??

?

0

??n

? 2 ?

(34)

where int[x] is the integer part of x, we see that we can rewrite (32) as

1

? I

?(1

?e

?qT)

?

Γ

1

??2

?

[sin

?n

??cos

?n]

?e

?(qT

??)[Ac

sin( ?n

?e

??? ?

? Ie

? ?](35)

where c

either zero or two roots, and if it has two, they are destroyed in a saddle–node bifurcation as

a parameter is varied. These saddle–node bifurcations mark the edge of the q : 1 tongue in

parameter space, and a q : 1 orbit can only exist within such a tongue.

Combining the two terms involving

?is the smallest root of (31). Regarding

?nas a variable, equation (35) typically has

?n, equation (35) can be rewritten as

1

? I

?e

??qT[Ac

?e

????

? Ie

??]

? (1

?e

?qT)

?

Γ

?1

??2

?

?tan

?1

?)(36)

The saddle–node bifurcations occur when

?n

???2

?tan

?1

?

or

?n

? 3 ??2

?tan

?1

?

(37)

at which point the right hand side of (36) is

Γ(1

?e

?qT)

?1

??2

or

?Γ(1

? e

?qT)

?1

??2

(38)

13

Page 14

respectively. Using this and rearranging (36) we see that the two values of I between which

q : 1 orbits exist are

I

?1

? Ac

?e

????e

??qT

1

?e

?qT

?

Γ

?1

??2

(39)

As an example, in Figure 8 we show the boundaries of the 1 : 1 tongue, calculated using (39).

This expression only gives the region in parameter space where such orbits exist, but says

nothing about their stability. For

is stable between the saddle–node bifurcations that mark the edges of the tongues, but for

? less than

? 6 ?5, there always appears to be an orbit which

? greater than

curve in Figure 8). This is a Hopf bifurcation of a periodic orbit in the continuous system (1)-

(2) and corresponds to the creation of a 2–torus. This bifurcation is subcritical [10, 24], and

one consequence of that is the bistability of the periodically–forced system, at least in some

neighborhood to the left of this dashed line, as shown in Figure 9. In this Figure we show the

? 6 ?5, the 1 : 1 orbit can lose stability through a Hopf bifurcation (dashed

behavior for 6 ?8

The coexistence of at least two attractors (one of which is the 1:1 locked orbit) for some range

of

6–dimensional model [26].

(To find the curve of Hopf bifurcations, 1:1 phase–locked solutions of (26)-(28) were found

by solving

??? 7 ?3 when I

? 1 ?22, i.e. a cut through the Hopf bifurcation in Figure 8.

? values is clearly seen. This bistability has also been observed in the periodically forced

( ??2 ?) ??? f( ????( ??2 ?) ???c) and

c

? h( ????( ??2 ?) ???c)(40)

for

are given by (26) and (28), respectively. The stability of these solutions was then determined

by evaluating the eigenvalues of a numerically–determined approximation of the Jacobian

of (26)-(28) at the corresponding points.)

This process of finding the boundaries of resonance tongues can be carried out for other

frequencyratios,buttheresultingequationsare morecomplicated. Notethatthisprocedureis

not affected by the presence of the periodic

ilar observation was made in [49], where the authors studied a periodically forced Fitzhugh–

Nagumo system as the underlying dynamics changed from excitable to oscillatory. Note also

that Figure 8 is by no means a complete description of the dynamics, as there are an infinite

number of resonance tongues in this parameter space, each labeled by the pair p : q, where p

and q are positive integers.

Recall from Figure 4 that when no forcing is applied, the system moves from periodic

firing to burstingat I

value of the DC component of the applied current at which the system starts to burst, e.g. at

?? [0 ?2 ?) (the phase of the forcing cycle at which the neuron fires) and c, where f and h

? bursting boundary in parameter space. A sim-

?1 ?22. ReferringtoFigure 8, we seethat periodicforcing can increase the

?? 6, as shown in Figure 10 (top). Here, the 1:1 tongue straddles the value of I at which

the unforced system starts to burst, and the system starts bursting at the boundary of the

tongue. Conversely, sinusoidal modulation can also decrease the value of the DC component

of the applied current at which the system moves to bursting, e.g. at

Figure 10 (bottom). Note that this decrease in threshold could not be predicted from looking

at Figure 8, since the transition into bursting for this value of

the 1:1 tongue. Thus the effective “burst threshold” can be either increased or decreased,

depending on the frequency of forcing, as was observed in [26]. This phenomenon of shifting

the effective threshold has not yet been observed with actual pyramidal cells, but should be

straight–forward to verify.

?? 7 ?15, as shown in

? does not involve leaving

14

Page 15

3.2Stochastic Resonance

Stochastic resonance is the phenomenonwhereby moderate amounts of noise, when added to

a systemthat has a subthresholdinput signal, cause the signal to be observable inthe system’s

output [12]. For small noise intensities the signal cannot be observed, as it is subthreshold,

and for high intensities the system’s output is swamped by the noise, so if the signal to noise

ratio at the output is plotted as a function of noise intensity, it will show a maximum at some

moderate value of noise intensity.

We have already derived a map, (26)-(28), for the sinusoidally forced system (1)-(2), and

we can use this to show stochastic resonance in (1)-(2), since it already incorporates a signal

— the sinusoid. One choice of modeling the effect of noise is to replace (28) by

cn

?1

? h(tn

?sn

?cn)

???n

(41)

where the

This noise could be due to, for example, the probabilistic nature of the opening and closing of

ion channels on the neuron being modeled, or the stochastic nature of synaptic transmission

from presynaptic neurons. The parameter

In order to quantify the signal to noise ratio, we need to choose which aspect of the sys-

tem (1)-(2) is to be considered as the output. As was done in [26], we use the high–frequency

“doublets” that occur at the end of a burst. These could be detected preferentially by, for

example, a synapse with facilitation that acts over a doublet ISI, but not over a typical ISI in-

volved in periodic firing. High frequency doublets have been linked to synchronization and

communication over long distances in the brain [11, 44]. It has also been suggestedthat bursts

(whose presence would be signaled by a doublet, in this example) rather than spikes may be

the important unit of information in neural communication [29].

Thus for a set of N iterates of (26), (27) and (41) we define the output signal to be

?nare chosen from a normal distribution with mean zero and standard deviation 1.

? controls the noise intensity.

f(t)

?∑

j

Æ(t

??j) (42)

where

??j

???ti: ti

?ti ?1

???N

i ?2

(43)

so f(t) consists of a sum of delta functions at the times of the second spike in a pair whose ISI

is less than

and taking the Fourier transform we have

(0 ?07 ?0 ?09)

noise intensity,

if the full system, (1)-(2), is simulated with Gaussian white noise added to either the dynam-

ics of V, or the dynamics of c (not shown). Thus the noise present in this particular neural

system may actually enhance the transmission of information from the fish’s environment to

its higher brain centers.

?. Passing f(t) through a Hanning window [35] defined over the interval [t1

?tN]

F( ?)

?1

2∑

j

ei

??j

?

1

?cos

?2 ?

tN

?j

? t1

??

(44)

and the powerspectrumis

(the results of 5 realizations have been averaged). A clear increase in power at the driv-

?F( ?) ?2. Anexample is shownin Figure11 for

??0 ?6 and N

?3000

ing frequency ( ?

appears at nearby frequencies. This is quantified in Figure 12, where we have plotted the

ratio of the power at the driving frequency (0.1) to the average of the power in the range

? 0 ?1) is seen as

? increases, but if it is increased too far, more power

?(0 ?11 ?0 ?13). This signal to noise ratio clearly increases and then decreases as the

?, increases, a characteristic of stochastic resonance. This behavior is also seen

15

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