Invariant Multiparameter Sensitivity ofOscillator Networks
Kenzaburo Fujiwara, Takuma Tanaka, and Kiyohiko Nakamura
Department of Computational Intelligence and Systems Science, Interdisciplinary
Graduate School of Science and Engineering,Tokyo Institute of Technology, Yokohama, 226-8502, Japankenzaburo.fujiwara@gmail.com

Abstract. The behavior of neuronal and other biological systems isdetermined by their parameter values. We introduce a new metric toquantify the sensitivity of output to parameter changes. This metric isreferred to as invariant multiparameter sensitivity (IMPS) because it
takes on the same value for a class of equivalent systems. As a simplification of neuronal membrane, we calculate, in parallel resistor circuits,
the values of IMPS and a previously studied metric of parameter sensitivity. Furthermore, we simulate phase oscillator models on complex
networks and clarify the property of IMPS.
Keywords: Parameter Sensitivity, Complex Network, Phase Oscillator,Synchronization.

1Introduction

A large number of mathematical models have been proposed in order to explain
complex phenomena in brain including learning, chaotic behavior and synchronization [1, 2]. Because these models have many parameters, it would be desirable
to know how changes of parameter values influence the output of a model. Information about the relationship between parameter changes and output of models
is indispensable in designing models, fitting parameters and understanding thedynamics of systems [3, 4].
In this paper, we investigate parameter sensitivity, that is, the response ofoutput to small changes of parameters. Parameter sensitivity has been intensively studied in circuit theory, particularly in resistor-capacitor networks [4–7].
In biochemical modeling, metrics of sensitivity are also used in quantifying robustness of systems [8]. In neuronal modeling and machine learning, it is important to estimate how sensitively output, such as firing rate and generalization
error, changes in response to small parameter changes.
Several metrics of parameter sensitivity have been proposed in previous studies. Single parameter sensitivity (SPS) allows us to quantify the output changes
in response to small change of a single parameter. Multiparameter sensitivity(MPS) is a generalization of SPS to multiple parameters [9]. MPS is defined asthe square root of the sum of the square of SPSs. However, as will be shown later,

2Kenzaburo Fujiwara, Takuma Tanaka, Kiyohiko Nakamura

AB

R

R’1
R’2VVIIFig. 1. Circuit A and circuit B are equivalent if R = R1′ R2′ /(R1′ + R2′ ).

MPS gives different values for such equivalent electric circuits as in Fig. 1(A)and Fig. 1(B).
We propose in this paper a new metric of sensitivity, which we call invariant
multiparameter sensitivity (IMPS). This paper is organized as follows. In section 2, we introduce sensitivity metrics previously proposed and define IMPS.
Then we derive basic properties of IMPS. In section 3, we examine propertiesof IMPS by applying it to a simple circuit. In section 4, we further investigate
IMPS for nonlinearly coupled oscillators. Since it was reported that networks inbrain are scale-free networks [10], in which the number of connections of eachvertex obeys a power-law distribution, we examine the system of oscillators on ascale-free network. In section 5, we summarize our results and discuss potential
applications.

2Parameter Sensitivity

Dynamical systems are expressed by first-order differential equationsẋ = F (t, x, p) ,(1)where t is time, x = [x1 , x2 , x3 , . . . , xn ] is the state variable vector and p =[p1 , p2 , p3 , . . . , pm ] is the parameter vector.
For the output q of the system, single parameter sensitivity for parameter pi ,which we call SPSi , is defined as
SPSi =∂ ln qpi ∂q=.
q ∂pi
∂ ln pi(2)

SPSi is the ratio of the change of output q to the change of parameter pi .
However, SPSi does not quantify sensitivity to the change of other parameters.
MPS, which is defined asm
X
2
SPS2i ,(3)MPS =i=1

Invariant Multiparameter Sensitivity of Oscillator Network3

is known as a metric to estimate sensitivity to the change of the whole parameterset of the system [9, 7]. As shown in the next section, MPS often gives different values for two equivalent models, and thereby MPS is not appropriate forcomparing sensitivities between models. Thus, we introduce a new metric, invariant multiparameter sensitivity (IMPS). IMPS is defined by the sum of absolutevalues of SPSs asmX|SPSi | .
(4)
IMPS =
i=1

IMPS gives the same values for equivalent models in many cases. Assuming thatq(p1 , p2 , p3 , . . . , pm ) is a homogeneous function of degree k and that SPSs inequation (4) have the same sign, we obtain
IMPS ==mXpi ∂q
i=1mXq ∂pipi ∂qq ∂pii=1= |k| ,where we used Euler’s theorem∂q∂q∂q
p1+ p2+ · · · + pm
= kq(p1 , p2 , p3 , . . . , pm ).
∂p1∂p2∂pm

(5)

(6)

Hence IMPS is constant. IMPS is invariant for all models satisfying the following conditions: (1) the outputs are expressed by homogeneous functions ofparameters; and (2) SPSs take on the same sign.

3Circuit Toy Models

In this section we examine circuit toy models. Consider that there is one resistor
R in a circuit as in Fig. 1(A). We denote the electric energy consumption by W
and the voltage of the voltage source by V . We assume that W is the output.
MPS of this circuit equals 1. The same current-voltage relationship as the circuit
shown in Fig. 1(A) can be realized by the circuits equivalent to it such as thatin Fig. 1(B) if R = R1′ R2′ /(R1′ + R2′ ). MPS of the circuit in Fig. 1(B) is given by
MPS2 =nX

SPS2i

i=1=

22  ′X
R ∂Wi

W ∂Ri′i=1
2 
2
R1′R2′+=
R1′ + R2′
R1′ + R2′
< 1.

(7)

4Kenzaburo Fujiwara, Takuma Tanaka, Kiyohiko Nakamura

Thus, MPS of the circuit in Fig. 1(B) is less than MPS of that in Fig. 1(A). Incontrast, IMPS of the circuit in Fig. 1(B) is given byIMPS =nX|SPSi |

i=1R1′R2′+R1′ + R2′
R1′ + R2′= 1,=(8)which equals IMPS of that in Fig. 1(A). It can be easily shown that IMPS isthe same for the energy consumption of the equivalent RC circuits, by which theelectric properties of neuronal membrane have been modeled [1].

4Nonlinear ModelIn this section, we investigate the IMPS of the system of phase oscillators on aBarabási–Albert network as an example of neuronal networks. Barabási–Albertmodel is the most thoroughly studied scale-free network model [11]. We generate
Barabási–Albert networks with average degree of 4. We start from 2 vertices andadd a vertex with 2 edges in each step until we have N vertices.
We assume that N oscillators are connected to each other by the adjacency
matrix ABA of a Barabási–Albert network. The dynamics of oscillator i aredescribed byNXdθiKij sin(θj − θi ),= ωi +dtj=1(9)where Kij is the (i, j)-element of the connection weight matrix defined by K =αABA and the natural frequencyP
ωi is drawn from the Gaussian distribution withNunit variance. We assume that i=1 ωi = 0 without loss of generality. Here, α
is the connection strength. We use the circular variance V of the oscillators
V =1−r =1−

1p 2
C + S2N(10)in thePphase-locked state asPan output, where r is the Kuramoto order parameter,NN
C = i=1 cos θi and S = i=1 sin θi .
In the phase-locked state, the right-hand side yi′ of equation (9) is 0, that is,y ′ = 0.

(11)

Here we derive the relationship between the connection weights and the phases
under the condition that equation (11) is satisfied. Assuming that ∆K is small,Invariant Multiparameter Sensitivity of Oscillator Network

5we obtainyi′ + ∆yi′ = ωi +NX(Kij + ∆Kij ) sin(θj + ∆θj − θi − ∆θi )j=1≈ ωi +NX(Kij + ∆Kij ) [sin(θj − θi ) + cos(θj − θi )(∆θj − ∆θi )] .

j=1

(12)

Subtracting yi′ from both sides yields∆yi′ ≈NX

Kij cos(θj − θi )(∆θj − ∆θi )

j=1+NX∆Kij [sin(θj − θi ) + cos(θj − θi )(∆θj − ∆θi )] .

(13)

j=1Thus we obtain∂yi′′≡ Jij,∂θj(sin(θm − θl )
∂yi′=
∂Klm
0(14)
i=l,
i 6= l(15)where( PN− s=1 Kis cos(θs − θi )′Jij=Kij cos(θj − θi )i=j.
i 6= j(16)J’ is of N − 1 rank, because Laplacian matrices of connected graphs are of N − 1rank [12]. Adding the same value to all θi ’s of a phase-locked solution results
in another phase-locked solution, and the latter cannot be distinguished fromthe former in terms of V . Thus we cannot determine the unique phase-locked
solution for this model. However, we can set the average phase to 0, which
will not ruin the generality of our argument, because
PN we are interested only incircular variance V of the oscillators. Assuming i=1 θi = 0, we can replace
equation (11) withy i ≡ ωi +NX

Kij sin(θj − θi ) +j=1NX

θj = 0.

(17)

j=1Hence ∂yi /∂θj can be derived as∂yi′= Jij+ 1 ≡ Jij .
∂θj(18)

6Kenzaburo Fujiwara, Takuma Tanaka, Kiyohiko Nakamura

J is full rank. Thus we haveNX∂θi(J−1 )ij δjl sin(θm − θl )=−∂Klmj=1= −(J−1 )il sin(θm − θl ).

(19)

Hence the derivative of V with respect to Klm is given by
2  P2 PNN∂+i=1 cos θi
i=1 sin θi−1/21∂V=−C2 + S2∂Klm2N∂Klm!
NNXX
1−1−1= 2sin θi (J )il sin(θm − θl ). (20)cos θi (J )il − CSN ri=1
i=1
From the above analysis, we numerically obtain IMPS as
IMPS =X|SPSlm | =hlmi

X Klm ∂V,V ∂Klm

(21)

hlmi

where hi is the summation over the connected oscillator pairs. In the initial state,
all phases are uniformly distributed. When α is sufficiently large, the oscillators
are phase locked as shown in Fig. 2. We calculate the IMPS for various valuesof α under phase-locked conditions.
2πθi

1.5ππ

0.5π
0012345TimeFig. 2. Synchronization of 1000 oscillators on a Barabási–Albert network with average
degree of 4. Phases of 4 out of 1000 oscillators are shown. The connection strength αis set to 2.

The IMPS of this model is shown in Fig. 3. IMPS gives similar values for
system size N = 1000 (Fig. 3A) and 10000 (Fig. 3B), whereas MPS exhibits
system-size dependency (Fig. 3C). Unlike the toy circuit model in section 3,

Invariant Multiparameter Sensitivity of Oscillator Network22 3 4 5 6 7 8α

N = 1000
N = 10000

MPSIMPS

2.1

0.2

2.2

2.2IMPSCBA72.1223 4 5 6 7 8α

0.102 3 4 5 6 7 8αFig. 3. IMPS of oscillator networks with N = 1000 (A) and N = 10000 (B). Panel Ashows the results of networks generated by 5 different random seeds. Panel C shows
the MPS for N = 1000 (dashed line) and the MPS for N = 10000 (solid line).

IMPS for this system of phase oscillators does not take on a constant value because of the nonlinearity of this system. If α is sufficiently large, the nonlinearityof sine coupling is ignorable. Therefore, as α increases, IMPS of this oscillator
system converges to 2 because the circular variance V of this model converges
to a homogeneous function of degree −2.

5Discussion

In this paper, we have reviewed the previously proposed metrics, SPS and MPS,
which quantify parameter sensitivity. We have formulated an improved metric,
IMPS. IMPS gives the same value for equivalent models in many cases. Thisgives IMPS a significant advantage over MPS, which gives different values forequivalent systems. In the analysis of the simple circuits, IMPS has given thesame value for equivalent parallel circuits. Then we have applied IMPS to nonlinear complex systems. As a first step for applying IMPS to neuronal systems,
we have used the phase oscillator model and the Barabási–Albert model because
those two models are widely used in the previous research [11, 13, 14].
Formerly, invariance of IMPS was reported only for RC network circuits [4–7]. In this paper, we have shown its invariance in a wider setting than previousstudies. In the system of phase oscillators, IMPS is not always invariant becauseof the nonlinearity. Our results suggest that IMPS is a metric reflecting both
structure and dynamics of the systems. Thus IMPS would allow us to estimate
the dynamics and excitability of individual neurons and synaptic connectivity
between neurons.
In a future work, it should be examined how structure and nonlinearity ofsystems are reflected in the value of IMPS. In particular, we will apply IMPS tothe system of neurons on a Watts–Strogatz small-world network [15]. Furthermore, the relation between IMPS and previously proposed network metrics, suchas cluster coefficient and average path length [15, 16], should be investigated. Theapplication of IMPS to the real neuronal networks is also of interest.

8Kenzaburo Fujiwara, Takuma Tanaka, Kiyohiko Nakamura

Acknowledgment
This work was supported by MEXT/JSPS KAKENHI Grant Numbers 24651184
and 25115710.

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