partial phase synchronization of neural populations due to

J Comput Neurosci (2008) 25:141–157DOI 10.1007/s10827-007-0069-z

Partial phase synchronization of neural populationsdue to random Poisson inputs

Per Danzl · Robert Hansen · Guillaume Bonnet ·Jeff Moehlis

Received: 15 October 2006 / Revised: 5 November 2007 / Accepted: 3 December 2007 / Published online: 28 December 2007© Springer Science + Business Media, LLC 2007

Abstract We show that populations of identical un-coupled neurons exhibit partial phase synchronizationwhen stimulated with independent, random unidirec-tional current spikes with interspike time intervalsdrawn from a Poisson distribution. We characterizethis partial synchronization using the phase distributionof the population, and consider analytical approxima-tions and numerical simulations of phase-reduced mod-els and the corresponding conductance-based modelsof typical Type I (Hindmarsh–Rose) and Type II(Hodgkin–Huxley) neurons, showing quantitativelyhow the extent of the partial phase synchronization de-pends on the magnitude and mean interspike frequencyof the stimulus. Furthermore, we present several simpleexamples that disprove the notion that phase synchronymust be strongly related to spike synchrony. Instead,the importance of partial phase synchrony is shown tolie in its influence on the response of the populationto stimulation, which we illustrate using first spike timehistograms.

Keywords Phase synchronization ·Population dynamics · Phase-reduced models

Action Editor: G. Bard Ermentrout

P. Danzl (B) · R. Hansen · J. MoehlisDepartment of Mechanical Engineering,University of California, Santa Barbara, CA 93106, USAe-mail: [email protected]

G. BonnetDepartment of Statistics and Applied Probability,University of California, Santa Barbara, CA 93106, USA

1 Introduction

Synchronized neural activity is believed to be importantfor various brain functions, including visual process-ing (Gray et al. 1989; Usrey and Reid 1999), odoridentification (Friedrich and Stopfer 2001), signal en-coding (Stevens and Zador 1998), cortical process-ing (Singer 1993), learning (Stent 1973), and memory(Klimesch 1996). It can also be detrimental. For exam-ple, resting tremor in patients with Parkinson’s diseasehas been linked to synchronization of a cluster of neu-rons in the thalamus and basal ganglia (Pare et al. 1990).Similarly, essential tremor and epileptic seizures arecommonly associated with synchronously firing neu-rons (Elble and Koller 1990; Traub et al. 1989), whichcan become (partially or fully) synchronized due tocoupling and/or stimulation by common inputs.

Even common inputs that are random or noisy canlead to synchronization (Goldobin and Pikovsky 2005a,b; Nakao et al. 2005). This applies to a broad rangeneuron models (Ritt 2003; Teramae and Tanaka 2004),with little constraint on intrinsic properties. It hasalso been shown experimentally that some neurons, inparticular olfactory bulb mitral cells, can synchronizein this manner in vitro (Galán et al. 2006). This isrelevant to spike timing reliability experiments, whichfound that repeated injection of the same fluctuatingcurrent into a single cortical neuron leads to a morereproducible spiking pattern than injection of a con-stant current (Mainen and Sejnowski 1995). Indeed,an experiment in which multiple uncoupled neuronsare subjected to a common input is equivalent to anexperiment in which a single neuron is subjected tothe same input over multiple trials, as in Mainen and

142 J Comput Neurosci (2008) 25:141–157

Sejnowski (1995), so that spike timing reliability can beviewed as synchronization across trials.

Typically, as in the references cited above, synchronyin the context of neuroscience is discussed in terms ofsynchronization of action potentials (spikes). Synchro-nization of spike times is a natural way to quantifythe dynamic behavior of a population of neurons, sincetypically the only observable quantities are voltages.There is, however, another form of synchrony that canplay an important role in the dynamic response of apopulation of oscillatory neurons to stimulus.

In this paper, we consider partial phase synchroniza-tion, a characteristic that provides information aboutthe dynamical state of a population of oscillatory neu-rons not easily obtained by studying spike synchronyalone. The concept of partial phase synchronizationapplies to populations of oscillatory neurons, eachof which evolves in time according to dynamics canbe represented by a one-dimensional phase oscillator(a “simple clock”, in the terminology of Winfree 2001).Here, the phase of a neuron relates its state, in time,to the firing of an action potential (or other markerevent on its periodic orbit), as described for examplein Guckenheimer (1975) and Brown et al. (2004a). Thedynamical state of a population of phase oscillatorscan be characterized by the distribution of their phases(over the unit circle). Partial phase synchronizationrefers to the degree to which this phase distributionpossesses a single dominant mode, meaning that thereis a higher density of neurons with phases near this re-gion than anywhere else on the circle. Complete phasesynchronization is the limiting case of partial phasesynchronization where all the neurons have exactly thesame phase, which yields a phase distribution in theform of a Dirac delta function.

After specifying the neuron and stimulus modelswe consider, we present an intuitive description ofthe mechanism by which partial phase synchronizationcan occur. We then consider a case in which partialphase synchronization occurs in a population of iden-tical uncoupled neurons that each receive different (un-correlated) spike trains with interspike time intervalsdrawn from a Poisson distribution, which could rep-resent, for example, the background activity of otherneurons (Softky and Koch 1993). We then presenta detailed theory for calculating the long-time phasedistribution, the results of which are compared withsimulation data for populations of phase oscillatorsand conductance-based neuron models. Two differentneural models are studied: Hindmarsh–Rose (Rose andHindmarsh 1989), a prototypical Type I neuron, andHodgkin–Huxley (Hodgkin and Huxley 1952), a proto-typical Type II neuron (Rinzel and Ermentrout 1998).

These models are considered for parameter regions inwhich the neurons spike periodically in time. We showthat the simulation results for a population of phaseoscillators very closely match those for a populationof conductance-based neuron models exposed to thesame type of stimulus, and both sets of simulation datayield long-time phase distributions that are very similarto the theoretical predictions. This verifies that theconcept of phase synchronization, although developedbased on phase oscillators, is an effective and accuratetool for modeling populations of conductance-basedneuron models. We then exploit the computationalefficiency of simulating phase oscillator models in or-der to map how the degree of phase synchronizationdepends on the magnitude and mean spike frequency ofthe stimulus.

The balance of the paper will be devoted to dis-cussing the practical implications of partial phase syn-chrony to the response of the population. We willdevelop several simple examples that illustrate someof the complex relationships between phase synchro-nization and spike synchronization. We stress thatphase synchronization does not necessarily imply spikesynchronization, and spike synchronization guaranteesphase synchronization only in the weakest sense. Thetwo phenomena are distinct, and both are importantdynamical characteristics of a neural population.

2 Methods

2.1 Models

2.1.1 Conductance-based models

We consider a population of N identical neurons withdynamics governed by a conductance-based model ofthe form

CVi = Ig(Vi, ni) + Ib + Ii(t), (1)

ni = G(Vi, ni), (2)

for i = 1, · · · , N. For the ith neuron, Vi ∈ R is the volt-age across the membrane, ni ∈ R

m[0,1] is the vector of

gating variables, C ∈ R+ is the membrane capacitance,

Ig : R × Rm → R is the sum of the membrane currents,

Ib ∈ R is a constant baseline current, and Ii : R → R isthe current stimulus.

We choose prototypical systems to represent twocommon types of neurons (Rinzel and Ermentrout1998). As a Type I neuron model, we consider theHindmarsh–Rose equations (Rose and Hindmarsh

J Comput Neurosci (2008) 25:141–157 143

1989), which represent a reduction of the Connormodel for crustacean axons (Connor et al. 1977):

V = [Ib + Ii(t) − gNam∞(V)3(−3(q − Bb∞(V))

+ 0.85)(V − VNa) − gKq(V − VK)

− gL(V − VL)]/C,

q = (q∞(V) − q)/τq(V),

q∞(V) = n∞(V)4 + Bb∞(V),

b∞(V) = (1/(1 + exp(γb (V + 53.3))))4,

m∞(V) = αm(V)/(αm(V) + βm(V)),

n∞(V) = αn(V)/(αn(V) + βn(V)),

τq(V) = (τb (V) + τn(V))/2,

τn(V) = Tn/(αn(V) + βn(V)),

τb (V) = Tb (1.24 + 2.678/

(1 + exp((V + 50)/16.027))),

αn(V) = 0.01(V + 45.7)/(1 − exp(−(V + 45.7)/10)),

αm(V) = 0.1(V + 29.7)/(1 − exp(−(V + 29.7)/10)),

βn(V) = 0.125 exp(−(V + 55.7)/80),

βm(V) = 4 exp(−(V + 54.7)/18).

VNa = 55 mV, VK = −72 mV,

VL = −17 mV, gNa = 120 mS/cm2,

gK = 20 mS/cm2, gL = 0.3 mS/cm2,

gA = 47.7 mS/cm2,

C = 1 μF/cm2, γb = 0.069 mV−1,

Tb = 1 ms, Tn = 0.52 ms, B = 1.26.

As a Type II neuron model, we consider the Hodgkin–Huxley equations, which model the squid Loligo giantaxon (Hodgkin and Huxley 1952):

dV/dt = (Ib + Ii(t) − gNah(V − VNa)m3

− gK(V − VK)n4 − gL(V − VL))/C,

dm/dt = am(V)(1 − m) − b m(V)m,

dh/dt = ah(V)(1 − h) − b h(V)h,

dn/dt = an(V)(1 − n) − b n(V)n,

αm(V) = 0.1(V + 40)/(1 − exp(−(V + 40)/10)),

βm(V) = 4 exp(−(V + 65)/18),

αh(V) = 0.07 exp(−(V + 65)/20),

βh(V) = 1/(1 + exp(−(V + 35)/10)),

αn(V) = 0.01(V + 55)/(1 − exp(−(V + 55)/10)),

βn(V) = 0.125 exp(−(V + 65)/80).

VNa = 50 mV, VK = −77 mV, VL = −54.4 mV,

gNa = 120 mS/cm2, gK = 36 mS/cm2

gL = 0.3 mS/cm2, C = 1 μF/cm2.

We hope the use of these prototypical models willprovide the reader with some familiarity and intuition.

2.1.2 Phase models

We choose baseline current, Ib , values such that eachneuron’s only attractor is a stable periodic orbit: Ib = 5mA for the Hindmarsh–Rose model, and Ib = 10 mAfor the Hodgkin–Huxley model. Since each system hasa stable periodic orbit at the prescribed Ib value, it isuseful to map the system to phase coordinates (Brownet al. 2004a; Guckenheimer 1975; Kuramoto 1984;Winfree 1974, 2001). Following Guckenheimer (1975),there exists a homeomorphism h: R × R

m → S1 map-

ping any point on the periodic orbit γ to a unique pointthe unit circle θ ∈ [0,2π). Furthermore, in a neighbor-hood of the periodic orbit, the phases are determinedby isochrons, whose level-sets are the sets of all initialconditions for which the distance between trajectoriesstarting on the same isochron goes to zero as t → ∞(Josic et al. 2006). By convention, when θi = 0 the ithneuron fires. In the limit of small perturbations, thestimulus spikes serve to nudge the state slightly off theperiodic orbit. In this way, the state may be movedonto a different isochron resulting in a difference inphase, with a magnitude and direction determined bythe phase response curve (PRC). This reduction en-ables the dynamics of an (m + 1)-dimensional ordinarydifferential equation (ODE) neuron model to be rep-resented by the evolution of a scalar phase variable. Arecent discussion of this framework was communicatedin Gutkin et al. (2005).

We can therefore model our neural population bya set of one-dimensional phase models - one for eachneuron. In the presence of stimuli Ii(t), this phase re-duction yields the following uncoupled N-dimensionalsystem of equations for the phases θi of a population ofN identical neurons (Brown et al. 2004a; Gutkin et al.2005):

dθi

dt= ω + ZV(θi)

CIi(t), i = 1, · · · N. (3)

Here θi ∈ [0, 2π), ω = 2π/T where T is the period ofthe periodic orbit, and Z(θ) is the PRC. Of particular


Fig. 1 Phase response curve approximations the for Hindmarsh–Rose model with Ib = 5 mA (solid) and the Hodgkin–Huxleymodel with Ib = 10 mA (dashed)

interest are perturbations in the voltage direction,i.e. ZV(θ) = ∂θ

∂V . The software package XPPAUT(Ermentrout 2002) was used to numerically calculatePRCs for the Hindmarsh–Rose and Hodgkin–Huxleymodels. For computational convenience, we use ap-proximations to these PRCs shown in Fig. 1: theHindmarsh–Rose PRC is approximated by a curve-fitof the form ZV(θ) ≈ K

ω(1 − cos(θ)) where K ≈ 0.0036

(mV ms)−1 for Ib = 5 mA (Brown et al. 2004a, b),cf. Ermentrout (1996); the Hodgkin–Huxley PRC isapproximated as a Fourier series with 21 terms.

2.1.3 Stimulus model

To model the independent random stimuli, we supposethat each neuron receives δ-function current inputs ofstrength I at times determined by a Poisson processwith mean frequency α:

Ii(t) = I∑

k

δ(t − tk

i

), (4)

where tki is the time of the kth input to the ith neuron.

The times of these inputs are determined by drawingthe interspike intervals from the distribution

p(τ ; α) = αe−ατ . (5)

We emphasize that the neurons in our models are notcoupled to each other, and that each one receives thesame baseline current Ib but a different current stimulusIi(t).

In our analysis, we are interested in determininghow the population-level behavior depends on α and I.Softky and Koch (1993) found that the mean excitatory

spike frequency in certain brain neurons is about 83 Hzor α = .083 ≈ 0.1 spikes/ms. This falls at the high endof the gamma-rhythm range (Kopell 2000), and setsthe scale for biologically relevant α values. We baseour I scaling on a unit 1 mA, so that a spike givesan instantaneous voltage change of order I/C = 1 mV.Since the dynamic range of a periodically firing neuronis approximately 100 mV, this corresponds to a smallperturbation in the depolarizing direction (Tass 2000).

2.2 Mechanism for partial phase synchronization

2.2.1 Intuitive description

We take partial phase synchronization to mean thatthere is a higher probability of a neuron having a cer-tain range of phases than another range of equal size.The following illustrates qualitatively how partial phasesynchronization can occur for a population of neuronssubjected to the previously described stimuli. Take, forexample, the PRC ZV(θ) = 1 − cos(θ). Recall the timeevolution of a neuron is governed by Eq. (3). Nowimagine a set of four such neurons on the unit circlewith this PRC starting at {θ1, θ2, θ3, θ4} = {0, π

2 , π, 3π2 }.

These are shown as black markers on Fig. 2(a) and arelabeled according to their ith indices. Figure 2(a) alsoshows the phase of the four neurons in the absenceof stimuli after some time interval �t has elapsed:each neuron has advanced in phase by �θ = ω�t topositions indicated with open markers, labeled i′. Thisbehavior is termed drift and is simply due to the naturalfrequency of the neurons. If each of the neurons wasexposed to a unit stimulus during this same time inter-val �t, their phases advance additionally, as determinedby the PRC, by �θ = ZV(θ) I/C (if the PRC containsnegative values, the phase can also be retarded). Theposition of each of the neurons after drift and stimulusis indicated by striped markers labeled by i′′. Noticethat the position of neuron 1 (starting at θ1 = 0) withstimulus (1′′) is the same as without stimulus (1′). Thisis because ZV(0) = 0 for this example.

If all four neurons are subjected to stimulus, it isapparent from the i′′ locations in Fig. 2(a) that the distri-bution of neurons has changed. What began as uniformhas become asymmetric. The situation becomes moreapparent if many more neurons are displayed, as inFig. 2(b). Here we see a definite “bunching” aroundθ = 0. Many neurons have similar phases, hence phasesynchronization.

The above description assumed each neuron re-ceives the same input. Conceptually, one can envisiona similar argument if all neurons receive unidirectional


(a) (b)Fig. 2 Initial phases as black markers labeled i, phase after drift as open markers labeled i′, and phase after drift and stimulus as stripedmarkers labeled i′′ (a). Distribution of many neurons after exposure to stimulus (b)

independent Poisson-type stimuli with the same statis-tics, i.e. mean interspike frequency α and magnitude I.

2.2.2 Theoretical development

Suppose that each neuron in a population receivedthe same non-random stimulus I(t). For example, I(t)might be a step function stimulus (Brown et al. 2004a)or a sinusoidal stimulus. We are interested in the prob-ability that a neuron will have a phase between θ andθ + dθ at time t, given by ρ(θ, t)dθ , where ρ(θ, t) is theprobability distribution function for the population ofneurons. The phase synchronization can be representedby the shape of ρ(θ, t) over θ ∈ [0, 2π) at time t. The fol-lowing partial differential equation can be derived forN → ∞ for such a population of neurons (e.g., Brownet al. 2004a):

∂ρ

∂t= − ∂

∂θ

[(ω + Z (θ)I(t)/C)ρ(θ, t)

]. (6)

For the present problem with independent randominputs Ii(t), the situation is more complicated. We pro-ceed by deriving an expansion of the density evolutionthat is conducive to perturbation methods consistentwith the neuron models and random inputs under con-sideration. For the values of mean spike frequency, α,and neuron natural firing frequency, ω, used in thispaper, it is reasonable to consider the ratio ω/α asO(1) = O(ε0) for the Hodgkin–Huxley model and O(ε)

for the Hindmarsh–Rose model, where ε is a smallparameter.

2.2.3 Kramers–Moyal expansion

The probability distribution ρ(θ, t) obeys the Kramers–Moyal expansion (Coffey et al. 2004)

∂ρ(θ, t)∂t

=∞∑

n=1

(− ∂

∂θ

)n

[D(n)(θ)ρ(θ, t)], (7)

where

D(n)(θ0) = 1

n! lim�t→0

E[(θ(�t) − θ0)n]

�t. (8)

Here E denotes the expected value, and θ(0) = θ0 forthe realizations used to calculate D(n)(θ0). Now

θ(�t) = θ0 + ω�t + �θ,

where �θ is the total jump in θ due to the randomPoisson inputs in [0, �t). For sufficiently small �t,we expect that either no inputs occur, in which case�θ = 0, or one input occurs, giving

�θ ≈ I Z (θ0)/C ≡ (�θ)jump. (9)

For n = 1 and sufficiently small �t, we have

E[θ(�t) − θ0] = ω�t + E[�θ |input]×p(input) + E[�θ |no input]×p(no input). (10)

Using

E[�θ |input] = (�θ)jump, E[�θ |no input] = 0,

p(input) = α�t, p(no input) = 1 − α�t


gives

E[θ(�t) − θ0] = ω�t + α�tZ (θ0) I

C, (11)

so that

D(1)(θ0) = ω + αZ (θ0) I

C. (12)

Similarly for n > 1,

E[(θ(�t) − θ0)n] = E[(ω�t + �θ)n]

= α�t

(I Z (θ0)

C

)n

+ O((�t)2), (13)

so that

D(n)(θ0) = α

n!

(I Z (θ0)

C

)n

n > 1. (14)

Finally, letting

Z (θ) = Zdψ(θ), ε = Zd IC

, (15)

where Zd and ε are nondimensional, we obtain thefollowing equation for the steady (∂/∂t = 0) densityρs(θ):

0 = −ω

α

dρs

dθ+

∞∑

n=1

εn (−1)n

n!(

ddθ

)n

[(ψ(θ))nρs(θ)]. (16)

We now consider the small ε limit, corresponding torandom inputs with small amplitude. We substitute

ρs(θ) = ρ0(θ) + ερ1(θ) + ε2ρ2(θ) + · · · (17)

into Eq. (16), and to solve at successive orders of ε.We divide this into two cases based on the relative sizeof ω/α.

2.2.4 Hodgkin–Huxley: ωα

= O(1)

This case applies when the ratio of natural frequency tomean spike frequency is of order 1, which correspondsto the Hodgkin–Huxley model in our analysis.

At O(ε0), we find that

dρ0

dθ= 0 ⇒ ρ0 = constant = 1

2π, (18)

where the value of the constant follows from thenormalization

1 =∫ 2π

0ρ0dθ. (19)

At O(ε), we obtain

−ω

α

dρ1

dθ− dψ

dθρ0 = 0, (20)

which has solution

ρ1(θ) = − α

2πωψ(θ) + k1. (21)

The value of the constant k1 is determined by thenormalization condition

∫ 2π

0ρ1(θ)dθ = 0 ⇒ k1 = α

(2π)2ω

∫ 2π

0ψ(θ)dθ.

(22)

Thus, to O(ε),

ρs(θ) ≈ 1

2π+ α I

2πωC

(−Z (θ) + 1

2π

∫ 2π

0Z (θ)dθ

).

(23)

At O(ε2), we obtain

−ω

α

dρ2

dθ− d

dθ[ψ(θ)ρ1(θ)] + ρ0

2

d2

dθ2[(ψ(θ))2] = 0. (24)

This has solution

ρ2(θ) = −α

ωψ(θ)ρ1(θ) + α

4πω

ddθ

[(ψ(θ))2] + k2. (25)

Using the normalization condition∫ 2π

0ρ2(θ)dθ = 0, (26)

we obtain the following approximation for ρs(θ) accu-rate to O(ε2):

ρs(θ) ≈ 1

2π+ α I

2πωC

(−Z (θ) + 1

2π

∫ 2π

0Z (θ)dθ

)

+ α2 I2

2πω2C2

{[Z (θ)]2 − 1

2πZ (θ)

∫ 2π

0Z (θ)dθ

}

+ α I2

4πωC2

ddθ

[(Z (θ))2] + k′2, (27)

where

k′2 = 1

2π

∫ 2π

0

{α IωC

Z (θ)

×[

α I2πωC

(−Z (θ)+ 1

2π

∫ 2π

0Z (θ)dθ

)]

− α

4πω

(IC

)2d

dθ[(Z (θ))2]

}dθ. (28)

The truncation at O(ε2) is equivalent to formulating theproblem in terms of a Fokker–Plank equation, the sameform of which is known as the diffusion equation.


2.2.5 Hindmarsh–Rose: ωα

= O(ε)

This case applies when the natural frequncy is muchsmaller than the mean spike frequency, which corre-sponds to the Hindmarsh–Rose model in the regimestudied in this paper.

We letω

α= εω, (29)

where ω = O(1). At O(ε) we obtain

ωdρ0

dθ= − d

dθ[ψ(θ)ρ0(θ)]. (30)

This has solution

ρ0(θ) = c0

ω + α IC Z (θ)

, (31)

where

1 =∫ 2π

0ρ0(θ)dθ ⇒ c0 =

(∫ 2π

0

dθ

ω + α IC Z (θ)

)−1

.

(32)

A useful interpretation for this result is that, on aver-age, the current α I enters each neuron during everytime unit. If such current came in uniformly, so thatI(t) = α I, Eq. (6) would have a steady distributiongiven by Eq. (31).

At O(ε2) we obtain

ωdρ1

dθ= − d

dθ[ψ(θ)ρ1(θ)] + 1

2

d2

dθ2[(ψ(θ))2ρ0(θ)]. (33)

This has solution

ρ1(θ) = α I Zd

2C

ddθ

[(ψ(θ))2ρ0(θ)

] + c1

ω + α IC Z (θ)

. (34)

Using the normalization condition

0 =∫ 2π

0ρ1dθ, (35)

we obtain the following approximation for ρs(θ) accu-rate to O(ε):

ρs(θ) ≈ c0

ω + α IC Z (θ)

+ α I2

2C2

ddθ

[(Z (θ))2ρ0(θ)

] + c′1

ω + α IC Z (θ)

,

(36)

where

c′1 = −c0

∫ 2π

0

ddθ

[(Z (θ))2ρ0(θ)

]

ω + α IC Z (θ)

dθ. (37)

A different approach for the derivation of the steadyprobability distribution for a related problem is givenin Nakao et al. (2005), which is primarily concernedwith showing phase synchronization in the case in whichall Ii(t) in Eq. (3) are identical. Here the dynamics arereduced to a random phase map, and the evolution ofthe density associated with this map is described bya generalized Frobenius–Perron equation. The steadydistribution can be found numerically, or analytically incertain limits. We find that our theory described abovegives good agreement with our numerical results, andprovides a straightforward alternative to the approachof Nakao et al. (2005).

2.3 Simulation methods

2.3.1 Conductance-based models

A numerical routine was constructed to simulatelarge populations of neurons described by (m + 1)-dimensional conductance-based models. Since the neu-rons are not coupled, the simulations can be conductedon a neuron-by-neuron basis. A second-order Runge–Kutta method with small fixed time step �t was used forO((�t)2) accuracy and compatibility with the Poisson-distributed stimuli. In our numerical models, we ap-proximate the δ-function by a rectangular spike ofduration �t and magnitude 1/�t.

To obtain a uniform initial phase distribution acrossthe population, the (m + 1)-dimensional periodic orbitis calculated at run-time as a set of (m + 2)-tuples ofthe states and time. The time is then scaled to [0, 2π),and initial states are assigned by drawing a scalar θ0

from a uniform random number generator and usingthe periodic orbit to interpolate the initial data. Oncethe initial condition has been set, the system is inte-grated. Periodically throughout the time evolution ofeach neuron, the program computes its phase using aroutine which will presently be described, yielding asampled time-series of the neuron’s phase even thoughthe equations are not in phase-reduced form.

Intuitively, the phase should be related to the time itwould take the neuron to fire, beginning at its presentstate, in the absence of stimuli. Computationally, someadditional care is required. We use the Hodgkin–Huxley equations as an example to illustrate some ofthe computational complexities. For the Ib values con-sidered, the vector field of this system contains a spiralsource near the periodic orbit (Rinzel and Miller 1980),an example of a phase singularity (Winfree 2001).Therefore, it is possible for a trajectory on the periodicorbit passing near this source to be pushed arbitrarily


close to it by stimulus. A neuron with a state verynear the spiral source may take a significant amountof time to evolve back to a neighborhood close to theperiodic orbit, especially if the (positive) real parts ofthe unstable eigenvalues are small. This situation isextremely rare for the models and parameter rangeswe consider. In fact, it did not occur for any of thesimulations which generated the data presented in thispaper. However, to be useful in a general setting, thephase calculation scheme must be able to identify thiscase and alert the program that a phase singularity hasbeen reached. In most settings dealing with behavior oflarge populations, these cases can be dismissed withoutloss of statistical significance, since they appear to occuron a set of extremely small measure.

To compute phase, we use the current state vectoras the initial (t = 0) data for a new simulation which isrun in absence of stimulus. This simulation is integrateduntil two spikes are observed. Let the time it takesfor a spike (above a set voltage threshold) to occur beT1 (a large upper bound on the allowable T1 is usedto identify possible phase singularities). Let the timeat which the next spike occurs be T2. The interspikeinterval between the two is then Tint = T2 − T1. Thenphase can be computed as

θ = 2π

(1 − T1modTint

Tint

). (38)

An important component of this algorithm is itsability to correctly identify a spike. We use a selectionof logical checks which test the voltage time seriesfor maxima above a set threshold, and have a pre-scribed minimum interspike time interval. Obviously,some knowledge of the spike magnitude and period isessential, but this is readily available since the com-putation of the periodic orbit at run-time provides allthe necessary information. Using this method, one cansimulate N neurons for P sample steps and receive, asoutput, an N × P matrix containing the discrete timeevolution of θi as the values in row i.

The above method provides an accurate and robustway of simulating and determining phase informationfor large populations of uncoupled neurons of arbi-trary dimension with a wide range of forcing functionsand environmental parameters. Not limited to smallperturbations, the utility of this method is constrainedonly by the relative stiffness of the neuron model andthe geometry of the basin of attraction of the periodicorbit. However, such flexibility comes at the price ofcomputational speed, which is strongly dependent on

the dimension of the neuron model and the type ofsolver used.

2.3.2 Phase-reduced models

We are interested in the population dynamics overas wide a (α, I) range as possible. In order to moreefficiently map the parameter space, we use the phase-reduced form of the neuron equations. We have imple-mented an algorithm which very efficiently simulatesEq. (3) for large N and for long times. It exploits thefact that Eq. (3) can be solved exactly for the time inter-vals for which no inputs are present. One gets times atwhich inputs occur by recognizing that the time intervalτ between subsequent inputs for an individual neu-ron can be obtained by sampling the distribution (5).When an input comes for the ith neuron, we instanta-neously let

θi → θi + ZV(θi) I/C.

Again, the uncoupled assumption of the populationallows each θi to be simulated independently, and theprogram creates an N × P output matrix of the discretetime evolution of θi.

3 Results

We simulate populations of uncoupled neurons eachsubjected to independent spike trains of set magnitude,I, with interspike intervals drawn from a Poisson distri-bution parameterized by mean spike frequency α. Ourresults include grayscale plots of ρ(θ, t) showing thetime evolution of the phase distribution of the popula-tion, time-averaged distribution curves, and mappingsof the distribution peak value and location as a functionof I and α. Comparisons are made between simulationsof the full conductance-based models, simulations ofthe phase models, and theoretical estimates.

For notational convenience, we will refer to theN × P output matrix as �. We remind the reader thatthe ith row of � represents the time-series values ofθi, the phase of neuron i, as it evolves from initialtime t0 to final time t f . While the differential equa-tions are solved using a small time step �t, we reportphase using a larger time step �p. This ensures thedisplayed data will not have spurious characteristicsdue to limitations of graphics resolution, and results ina dramatic performance increase in the simulations ofthe full conductance-based models. Therefore, the Pcolumns of � represent the (t f − t0)/�p sample points,plus a left-concatenated column of the initial phases, i.e.P = 1 + (t f − t0)/�p.


Table 1 Simulation parameters for Figs. 3 and 4.

Parameter Hindmarsh–Rose Hodgkin–Huxley

Ib (mA) 5 10α (spikes/ms) 0.1 1t f (ms) 1,000 100�p (ms) 1 0.1

3.1 Phase distribution dynamics

The phase density of the population at each samplepoint, ρ(θ, t) is computed from � by taking an appro-priately scaled histogram of the corresponding columnof �. For example, ρ(θ) at the kth sample point iscalculated by

ρ(θ, t = k�p) = hist(�ek, nbins)nbins

2π N, (39)

where hist(., .) is the standard histogram binning func-tion, nbins is the number of bins dividing the interval[0, 2π), and ek is an P × 1 vector of zeros with a 1 in thekth entry. The argument �ek is simply the kth column

of �, representing the phase of all the neurons at timet = (k + 1)�p, where k = 1 implies t = t0. The scalingused in Eq. (39) gives the normalizing condition

nbins∑

j=1

ρ

(θ = j

nbins2π, t

)�θ = 1, ∀t. (40)

where �θ is the bin size 2π/nbins. Equation (39) returnsa nbins × 1 vector discretization of ρ(θ, t).

Populations of 1,000 neurons were simulated, begin-ning with a uniform phase distribution at t0 = 0, andsubjected to random sequences of spikes with magni-tude I = 1 mA. An integration time step of �t = 0.01ms was used. Since the time scales of the Hindmarsh–Rose and Hodgkin–Huxley equations are significantlydifferent, separate simulation parameters are neces-sary, as given in Table 1.

Figure 3(a, b) shows our results for the simulationsof the full conductance-based models, to be comparedwith Fig. 3(c, d), which is the results from the phase-reduced model simulations. It is apparent that thephase reduction yields both qualitatively and quantita-tively similar results. Since the phase response curve

(a)

(c) (d)

(b)

Fig. 3 Simulation results, ρ(θ, t), for Hindmarsh–Rose (a), (c) and Hodgkin–Huxley (b), (d) (resp., conductance-based model, phase-reduced model)


(a) (b)Fig. 4 Comparison of theoretical distribution estimate withaveraged simulation data for both conductance-based and phase-reduced models for Hindmarsh–Rose (a) and Hodgkin–Huxley

(b). Solid lines represent full conductance-based model simula-tion data, dashed lines represent phase-reduced model simulationdata, and dotted lines represent the theoretical estimates

for the Hodgkin–Huxley system is of relatively smallmagnitude, we have used an α value of 1, rather than0.1, to more clearly illustrate the dynamic behavior ofthe phase distribution.

One notices in these figures a behavior that canbe described as “breathing”, i.e. there is an oscilla-tion (with period of approximately 200 ms for theHindmarsh–Rose model and approximately 15 ms forthe Hodgkin–Huxley model) about a mean distribu-tion. We found numerically that such oscillations persistfor long times of the order of 106 ms. It is worth notingthat this oscillatory behavior is exhibited in both thephase-reduced and conductance-based model simula-tions, suggesting that it may not be an artifact of thenumerics, but rather something inherent in the dynam-ics which is not predicted by the theoretical framework.For the purposes of our subsequent analysis, we viewthese oscillations as a secondary effect, to be pursuedin future work. To illustrate the average shape of theρ(θ, t) distribution curves, we average each simulationover the last 25% of the integration time interval,which filters the oscillatory effects of the “breathing”behavior.

Figure 4(a) shows that theoretical predictions matchthe numerical data very closely for the Hindmarsh–Rose system. The results for the Hodgkin–Huxley sys-tem are qualitatively similar, as illustrated in Fig. 4(b),but display a slight mismatch of the θ location of thedistribution peak.

3.2 Parametric study

The computationally efficient phase-reduced modelallows for a more complete mapping of the populationresponse in parameter space. As a characteriza-tion of the population response, we consider the mag-nitude, ρmax, and location, θmax, of the peak ofthe t → ∞ averaged probability distribution. As inSection 3.1, we consider populations of 1,000 neurons.The Hindmarsh–Rose neurons are simulated for 1,000ms (natural period ≈ 312 ms). The Hodgkin–Huxleyneurons are simulated for 100 ms (natural period ≈14.6 ms).

Figure 5 compares the phase model results to the the-oretical prediction using Eq. (36) for the Hindmarsh–Rose system over 0 < α ≤ 3 spikes/ms and 0 < I ≤ 5mA, for the ρmax characteristic. We find qualitative andreasonable quantitative agreement. For simulations ofthe full conductance-based model, simulations of thephase model, and theoretical predictions, the locationθmax of the probability distribution function peak ρmax

stays at θmax = 0.The plots in Fig. 6 show our results over the same

ranges of α and I for the Hodgkin–Huxley system.Figure 6(a) illustrates the more complex parameterdependence of ρmax. While the theory predicts thatθmax will be independent of α and I, Fig. 6(b) showsthat it is weakly dependent on them. Since our the-ory is based on the ratio ω/α being order ε, we have


(a) (b)Fig. 5 Numerical (a) and theoretical (b) ρmax over (α, I) parameter space for Hindmarsh–Rose model

(a) (b)

(c) (d)Fig. 6 Hodgkin–Huxley population-level response characteris-tics. Numerical ρmax over (α, I) parameter space (a). Numericalθmax over (α, I) parameter space (b). Comparison of numerical

results (c) to theoretical predictions (d) over the range of (α, I)for which the assumption ω/α = O(1) is reasonable


plotted the theoretical results over a reduced regionof α and I parameter space, as shown in Fig. 6(d) tobe compared with the simulation results reproducedover this reduced range in Fig. 6(c). We see that thetheoretical results qualitatively capture the trend ofthe data, but start to differ quantitatively as α and Iincrease. Importantly, for both the Hindmarsh–Roseand Hodgkin–Huxley models, we have shown that thedegree of synchronization, as measured by the peak ofthe probability distribution function of the phase, in-creases with both the size and frequency of the stimuli,and can become quite substantial.

4 Discussion

4.1 Relationship between partial phase synchronyand spike synchrony

The importance of studying partial phase synchroniza-tion lies not in its relationship to spike synchrony, butin its characterization of the dynamical state of thepopulation, which governs the response of the popula-tion to future stimulus. Before addressing the practicalimplications of partial phase synchrony, we will illus-trate some important points regarding the relationshipof phase synchrony and spike synchrony by discussingseveral simple examples. We will show that, in thegeneral case of a population of stimulated neurons, thedegree of partial phase synchronization is only weaklyrelated to spike synchrony.

In the following scenarios, we consider a populationof N identical uncoupled phase oscillators each obeyingthe following ODE:

θi = ω + Z (θ)

CIi(t).

In the absence of input, i.e. Ii(t) ≡ 0, there is a strongrelationship between phase synchrony and spike syn-chrony for populations of uncoupled identical oscilla-tors. If all oscillators have identical phase, it is obviousthat they will cross the 0/2π spiking threshold at identi-cal time. At the opposite extreme, if the distribution ofthe population is uniform, it follows that the spike timeswill be uniform in time (and thus desynchronized).

In the presence of input, coupling, or large distri-butions of natural frequency, this strong relationshipno longer holds. Since we have restricted our attentionto populations of identical uncoupled neurons in thispaper, we will develop several simple examples thatillustrate the how this relationship breaks down in thepresence of input stimulus.

Our primary concern is showing the mathemati-cal relationship between phase synchrony and spike

synchrony, so we will assume that there are no con-straints on the inputs we can use, and the oscillatorsthemselves are governed by phase response curves thatallow them to be controlled to achieve any prescribeddynamics:

θi = f (θi).

This assumption is not true in general for real neurons(since the phase response curves are often nonlinearand non-invertible). The question of whether popula-tion of neurons with a given phase response curve canbe driven to achieve specific dynamics when stimulatedby inputs from a restricted set is a control theoreticproblem and an aim of our future research. For the timebeing, we will follow our controllability assumption,which will allow us to develop examples that are simpleand easy to verify.

In order to quantify partial phase synchroniza-tion, we introduce the Kuramoto order parameter(Kuramoto 1984):

r(t)e√−1ψ(t) = 1

N

N∑

i=0

e√−1θi(t). (41)

In particular, we consider the magnitude of the orderparameter, |r(t)|, which ranges from zero for a uniformphase distribution to one for a completely synchronizedphase distribution (a Dirac delta function).

4.1.1 Example 1: phase synchrony withoutspike synchrony

Consider a population of N (uncoupled) phase oscil-lators, labeled θ1, . . . , θN , evolving according to thefollowing ODE:

θi = f (θi)

={

(π−�)ωN2π

if θi ∈ {[0, π − �)⋃ [π + �, 2π)

}

�ωNπ(N−2)

otherwise

(42)

with the following initial conditions:

θi(0) ={

0 for i = 1π − � + 2�(i−2)

N−2 for i �= 1(43)

This population will have uniform interspike intervals,i.e. no spike synchrony whatsoever, while possessingan order parameter |r| → 1 as � → 0 and N → ∞,which indicates an arbitrarily high degree of phasesynchronization.

What has been done here is to define a neighbor-hood around θ = π , with interval measure equal to2�, where the neurons move very slowly. Outside this


neighborhood the neurons advance very quickly. Wehave set up the initial conditions such that most of theoscillators (all but two) are in the small region wherethey move slowly. The one that starts at zero quicklymoves to the starting edge of the π -neighborhoodphase interval. During the same time interval, the lastoscillator, θN , advances quickly from the end of theπ -neighborhood phase interval around the circle to the0/2π spiking threshold. Also during this time interval,all the other neurons slowly advance in order along theπ -neighborhood phase interval. After the interspiketime interval, the neurons have rotated positions, butthe overall population distribution is identical to whereit started from, so after relabeling the oscillators, we areback to the starting configuration. Now if we make �

small, we increase the level of phase synchronization,but the firing continues to be completely uniform (de-synchronized). In fact, as � approaches zero, all theneurons except one will be arbitrarily close to π , so|r| → N−1

N = 1 − 1N . Furthermore, as we increase the

number of neurons, |r| → 1 asymptotically. The valueof |r| will, of course, never reach 1, but it can madearbitrarily close.

We illustrate this with a numerical example consist-ing of a set of N = 6 phase oscillators with naturalfrequency ω = 1. We set � = 0.5 so that the trajectories

will not be overly crowded in the π -neighborhood andthe plotted results can be easily understood. Figure 7(a)shows the trajectories generated by Eq. (42) alongwith the magnitude of the order parameter |r(t)|. Theinterspike time intervals here are uniformly equal toratio of the natural period T to the number of neuronsN, so there is no spike synchrony whatsoever. Yet themagnitude of the order parameter is at all times above0.5. This value can be driven arbitrarily close to oneby increasing the number of neurons and decreasing �.We note that all the phase trajectories are in solid black,because it is not important to be able to differentiatebetween them. Any set of trajectories following suchpaths, regardless of which neuron is on which trajectoryat a given time, will generate the same population-level results. This convention will be used in the nextexample, as well.

4.1.2 Example 2: spike synchrony with minimalphase synchrony

To further simplify the following example, we will as-sume that we can prescribe the phase trajectory forthe ith oscillator directly, rather than via an ODE,according to the following equation:

θi(t) =

⎧⎪⎪⎨

⎪⎪⎩

π+2π(i−1)

nεtmodT if tmodT ∈ [0, ε)

π+2π(i−1)

n if tmodT ∈ [ε, T − ε)

π+2π(i−1)

n + π+2π(n−i)nε

(tmodT − (T − ε)) if tmodT ∈ [T − ε, T)

(44)

The population of oscillators evolving according to Eq.(44) has complete spike synchrony, but the trajectorieshave been configured such that the distribution will benearly uniform except near the spike times, where thedistribution will collapse toward a Dirac delta distribu-tion, then expand again to be nearly uniform after thespike. Thus we have shown that, even in the presenceof complete spike synchrony, the phase synchrony canbe confined to intervals of arbitrarily small measure.To quantify this, we note that the integral of the orderparameter magnitude

R(t) ≡∫ t

0|r(t)|dt

can be made arbitrarily small by letting ε → 0. Thismeans that the phase synchrony of this populationis zero except on a set of arbitrarily small (but stillpositive) measure. We conclude that, although spike

synchrony must be accompanied by phase synchrony atthe instants of the spikes, we cannot say more abouttheir relationship without considering controllability ofthe oscillators with respect to possible trajectories.

We illustrate this point with a small population of 6oscillators with natural period T = 1. Figure 7(b) showsthe trajectories generated by Eq. (44) as well as theassociated |r(t)|. For clarity, we choose ε = 0.1. Themagnitude of the order parameter is zero everywhereexcept in an ε-neighborhood of the spike times. All ofthe oscillators spike together, which means that there iscomplete spiking synchrony. By reducing ε, we can con-fine the periodic phase synchrony to arbitrarily smalltime intervals centered at the spike time.

4.1.3 Poisson inputs

For a population of neurons subjected to indepen-dent Poisson inputs as described in Section 2.1.3, the


(a) (b)Fig. 7 Illustrations showing that partial phase synchrony andspike synchrony need not be strongly related. (a) shows thescenario from Example 1 where there is a high degree of phasesynchronization but no spike synchronization. (b) shows thescenario from Example 2 where there is complete spike synchro-

nization but only small time intervals of phase synchronization.The bottom figures show a quantification of phase synchrony, asrepresented by the magnitude of the Kuramoto order parameter.The results can be sharpened by increasing the number of neu-rons, and reducing the parameters � and ε

independence of the inputs implies that the spike timeswill be uncorrelated. It is intuitively clear from Eq. (41)and Fig. 4, the magnitude of the order parameter r canbe non-zero when there are such inputs.

From a mathematical standpoint, we can illustratethis by considering the continuum limit as N → ∞ andasymptotically as t → ∞,

|r(t)| →∣∣∣∣

∫ 2π

0ρs(θ)e

√−1θdθ

∣∣∣∣ ∼ π

2(ρmax − ρmin). (45)

The last relation follows from the fact that the absolutevalue of the integral is equal to the magnitude of thefirst Fourier component: when ρs is unimodal and nottoo sharply peaked, we expect that

ρs(θ) ∼ 1

2π+ 1

2(ρmax − ρmin) sin(θ − θ0)

for some θ0, from which the relation follows. The valuesof ρmax are precisely the results computed in Section 3.2and displayed in Figs. 5 and 6. We note in this casethat ρmin ≈ 1

π− ρmax ≤ 1

2π. The neurons subjected to

Poisson inputs thus have partial phase synchroniza-tion, with an order parameter increasing with boththe magnitude of the stimulus and the mean spikefrequency, although there is no spike synchronization.In the next section, we will discuss how a populationthat is partially phase-synchronized reacts differently to

stimulus than if its phases were uniformly distributed.This will illustrate the importance of considering phasesynchronization in addition to spike synchronizationwhen studying the dynamical properties of populationsof oscillatory neurons.

4.2 Effects of partial phase synchronization

The degree of partial phase synchronization affects themanner in which the population responds to stimulus,as can most clearly be seen by constructing a histogramof the first spike times of each neuron after receivinga step stimulus of 1 mA. For both the Hindmarsh–Rose and Hodgkin–Huxley systems, populations of1,000 phase models are initialized with uniform phasedistribution. We expose the population to the step inputand track when the next zero-crossing (firing) occurs.Figure 8(a) shows the first spike time histogram (FSTH)for Hindmarsh–Rose and Fig. 8(b) shows the FSTHfor Hodgkin–Huxley. To illustrate the effect of partialphase synchronization, we repeat the above simulationsstarting from the partially synchronized distributionshown in Fig. 4(a) and (b). The results are shown belowthe FSTH plots for the uniform cases, in Fig. 8(c) and(d). We see that for the Hindmarsh–Rose system, thepartial phase synchronization tends to flatten the FSTHsomewhat. For the Hodgkin–Huxley system, using the


(a) (b)

(c) (d)Fig. 8 First spike time histograms for Hindmarsh–Rose (a), (c)and Hodgkin–Huxley (b), (d). (a) and (b) begin from a uniformdistribution. (c) and (d) begin from the partially synchronized

distributions shown in Fig. 4. This analysis tracks the time of thefirst spike after the unit step-function stimulus is turned on. 1,000realizations are used

curve from the α = 1 case for consistency, we see avery pronounced change in shape of the FSTH. Allfour FSTH plots are tabulated over 1,000 realizations,in order to show the character of the distributions inde-pendent of the randomness inherent in any individualrealization. These results show that while partial phasesynchrony does not imply firing synchrony, it can play

an important role in determining how the populationresponds to a common stimulus.

4.3 Conclusion

We have shown that populations of identical uncou-pled neurons exhibit partial phase synchronization


when stimulated with independent unidirectional cur-rent spikes with interspike time intervals drawn froma Poisson distribution. We characterized this partialsynchronization by the phase distribution for the pop-ulation, using analytical approximations and numericalsimulations of phase-reduced models and conductance-based models of typical Type I (Hindmarsh–Rose) andType II (Hodgkin–Huxley) neurons. The results fromthe different approaches agree well with each other. Wefound that the degree of partial phase synchronization,as measured by the peak of the phase distribution, in-creases with both the size and frequency of the stimuli,and can become quite substantial.

We have shown that partial phase synchronization,a distinct phenomenon from spike synchronization, isan important consideration when using phase reducedmodels to infer dynamical characteristics of spikingneurons. Our results show that neural populations sub-jected to background activity from other neurons donot have a uniform distribution of phases, as is some-times assumed in simulation studies. We show thatsuch non-uniformity leads to different population-levelresponse to other stimuli, suggesting that noisy inputsmust be carefully incorporated into simulation studiesin order to obtain biologically realistic results.

The present study isolated the effect of randomstimuli by considering populations of identical uncou-pled neurons. It would be interesting to explore phasesynchronization due to noisy inputs for heterogeneousand/or coupled neural populations.

Acknowledgements This work was supported by the NationalScience Foundation grant NSF-0547606 and a University ofCalifornia Council on Research and Instructional ResourcesFaculty Research Grant. P.D. acknowledges an National ScienceFoundation IGERT Fellowship in Computational Science andEngineering. G.B. is partially supported by NIH Grant R01GM078993. J.M. acknowledges an Alfred P. Sloan ResearchFellowship in Mathematics. We thank the anonymous refereesfor helpful comments.

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