ARTICLE IN PRESS

Neurocomputing 72 (2008) 630–635

Contents lists available at ScienceDirect

Neurocomputing

0925-23

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/neucom

Letters

Population stochastic dynamics for synaptic depression

Wentao Huang �, Licheng Jiao, Jianhua Jia

Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education of China, Institute of Intelligent Information Processing,

Xidian University, Xi’an 710071, PR China

a r t i c l e i n f o

Article history:

Received 10 July 2007

Received in revised form

12 July 2008

Accepted 29 July 2008

Communicated by T. Heskesstochastic dynamics to analyze the population of neurons with synaptic depression in the stochastic

Available online 10 September 2008

Keywords:

Stochastic dynamics

Population responses

Synaptic depression

Time-dependent inputs

12/$ - see front matter & 2008 Elsevier B.V. A

016/j.neucom.2008.07.007

esponding author.

ail address: hwtsch@163.com (W. Huang).

a b s t r a c t

Synaptic transmission between neocortical neurons often shows activity-dependent synaptic depres-

sion. Moreover, it is revealed recently that the synaptic depression has a more active role in information

processing. The population density approach is a viable method to describe the large populations of

neurons and has generated considerable interest recently. In this paper, we use the population

environment. We have derived an evolution equation of the membrane potential density function with

synaptic depression, and obtain several formulas for analytic computing the response of instantaneous

fire rate. Through a technical analysis, we arrive at several significant conclusions, which show that the

background inputs and the spatial distribution of synapses and the spatial–temporal relation of inputs

play an important role in information processing.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

The synapses are highly dynamic and show use-dependentplasticity over a wide range of time scales, from millisecondsto hours to days. Experiments by Markram and Tsodyks [10] havesuggested that Hebbian pairing in cortical pyramidal neuronspotentiates or depresses the transmission of a subsequentpresynaptic spike train at steady-state depending on whetherthe spike train is of low frequency or high frequency, respectively.The frequency above which pairing induced a significant decreasein steady-state synaptic efficacy was as low as about 20 Hz andthis value depends on such synaptic properties as probability ofrelease and time constant of recovery from short-term synapticdepression. Many studies show that short-term synaptic plasticityplays important roles in information processing [2,1,3,6,11,13].Synaptic short-term depression is one of the most commonexpressions of plasticity.

In many areas of the brain neurons are organized in popula-tions of units with similar properties. Prominent examples arecolumns in the visual cortex and somatosensory, and poolsof motor neurons. Given a large number of neurons withinsuch a column or pool it is sensible to describe the mean activityof the neuronal population rather than the spiking of individualneurons. Each cubic millimeter of cortical tissue contains about105 neurons [8]. This impressive number also suggests that a

ll rights reserved.

description of neuronal dynamics in terms of a population activityis more appropriate than a description on the single-neuron level.Noise has an important impact on information processing of thenervous system in vivo. It is significant for us to study thestimulus-and-response behavior of neuronal populations, espe-cially to transients or time-dependent inputs in the noisyenvironment. It has received a lot of attention in recent years[5,4,7,12]. The present work is concerned with the collectivitydynamics of neuronal population with synaptic depression. First,we deduce a one-dimension Fokker–Planck (FP) equation viareducing the high-dimension FP equations. Then, we derive thestationary solution and the response of instantaneous fire ratefrom it. Finally, the models are analyzed and discussed in theoryand some conclusions are presented.

2. Models and methods

2.1. Single neuron models and density evolution equations

Our approach is based on the integrate-and-fire (IF) neurons.Due to its simplicity, it has become one of the canonical spikingrenewal models, since it represents one of the few neuronalmodels for which analytical calculations can be performed. Itdescribes basic sub-threshold electrical properties of the neuron.The population density based on the IF neuronal model is low-dimensional and thus can be computed efficiently, although theapproach could be generalized to other neuron models. It iscompletely characterized by its membrane potential below

ARTICLE IN PRESS

W. Huang et al. / Neurocomputing 72 (2008) 630–635 631

threshold. Details of the generation of an action potential abovethe threshold are ignored. Synaptic and external inputs aresummed until it reaches a threshold where a spike is emitted.The general form of the dynamics of the membrane potential v inIF model can be written as

tvdvðtÞ

dt¼ �vðtÞ þ SeðtÞ þ tv

XN

k¼1

JkðtÞdðt � tspk Þ, (1)

where 0pvp1, tv is the membrane time constant, SeðtÞ is anexternal current directly injected in the neuron, N is the number ofsynaptic connections, tsp

k is occurring time of the firing of apresynaptic neuron k and obeys a Poisson distribution with meanlk, JkðtÞ is the efficacy of synapse k. The transmembrane potential,v, has been normalized so that v ¼ 0 marks the rest state, andv ¼ 1 the threshold for firing. When the latter is achieved, v isreset to zero. JkðtÞ ¼ ADkðtÞ, where Að51Þ is a constant represent-ing the absolute synaptic efficacy corresponding to the maximalpostsynaptic response obtained if all the synaptic resources arereleased at once, and DkðtÞ act in accordance with complexdynamics rule. We use the phenomenological model by Tsodyksand Markram [11] to simulate short-term synaptic depression:

dDkðtÞ

dt¼ð1� DkðtÞÞ

td� UkDkðtÞdðt � tsp

k Þ, (2)

where Dk is a ‘depression’ variable, Dk 2 ½0;1�, td is the recoverytime constant, Uk is a constant determining the step decreasein Dk. Using the diffusion approximation [9], we can get from (1)and (2)

tvdvðtÞ

dt¼ �vðtÞ þ SeðtÞ þ tv

XN

k¼1

ADkðlk þffiffiffiffiffilk

pxkðtÞÞ,

dDkðtÞ

dt¼ð1� DkÞ

td� UkDkðlk þ

ffiffiffiffiffilk

pxkðtÞÞ. (3)

The FP equation of Eqs. (1)–(3) (see Appendix) is

qpðt; v;DÞ

qt¼ �

qqv

�vþ Kv

tvp

� ��XN

k¼1

qqDkðKDk

pÞ

�XN

k¼1

qqvqDk

ðlkAUkD2k pÞ

þ1

2

q2

qv2

XN

k¼1

lkA2D2kp

!þXN

k¼1

q2

qD2k

ðlkU2k D2

kpÞ

( ),

Kv ¼ Se þXN

k¼1

tvlkADk; KDk¼ð1� DkÞ

td� lkUkDk, (4)

where D ¼ ðD1;D2; . . . ;DNÞ, and

pðt; v;DÞ ¼ pdðt;DjvÞpvðt; vÞ;

Z 1

0pdðt;DjvÞdD ¼ 1. (5)

Because x1ðtÞ; x2ðtÞ; . . . ;xNðtÞ is uncorrelated, so we could assumethat D1;D2; . . . ;DN are uncorrelated. Then we have

pdðt;DjvÞ ¼YNk¼1

~pkdðt;DkjvÞ, (6)

where ~pkdðt;DkjvÞ is the conditional probability density. Because

every neuron receives large numbers of input of synapse (N is verylarge), and Dk does not depend on v and the influence of single Dk

for v is very small ðA51Þ. Then, we can assume

~pkdðt;DkjvÞ � pk

dðt;DkÞ. (7)

Substituting (5) into (4), we get

pd

qpv

qtþ pv

qpd

qt

¼ �qqv

�vþ Kv

tvpvpd

� ��XN

k¼1

pv

qqDkðKDk

pdÞ

�XN

k¼1

qqvqDk

ðAUkD2klkpvpdÞ

þ1

2

q2

qv2

XN

k¼1

lkA2D2kpvpd

!þXN

k¼1

q2

qD2k

ðlkU2k D2

kpvpdÞ

( ). (8)

Integrating Eq. (8) over D, we get

tvqpvðt; vÞ

qt¼ �

qqvð�vþ ~KvÞpvðt; vÞ þ

Qv

2

q2pvðt; vÞ

qv2, (9)

where

~Kv ¼

ZKvpd dD ¼ Se þ

XN

k¼1

tvlkAmk; Qv ¼XN

k¼1

tvlkA2gk,

mk ¼

ZDkpk

dðt;DkÞdDk; gk ¼

ZD2

kpkdðt;DkÞdDk, (10)

and pkdðt;DkÞ satisfies the following FP equation

qpkd

qt¼ �

qqDkðKDk

pkdÞ þ

1

2

q2

qD2k

ðU2kD2

klkpkdÞ. (11)

From (10) and (11), we can get

dmk

dt¼ �

1

tdþ Ulk

� �mk þ

1

td,

dgk

dt¼ �

2

tdþ ð2U � U2

Þlk

� �gk þ

2mk

td. (12)

Let

Jvðt; vÞ ¼�vþ ~Kv

tv

!pvðt; vÞ �

Qv

2tv

qpvðt; vÞ

qv,

rðtÞ ¼ Jvðt;1Þ, (13)

where Jvðt; vÞ is the probability flux of pv, rðtÞ is the fire rate. Theboundary conditions of Eq. (9) are

pvðt;1Þ ¼ 0;

Z 1

0pvðt; vÞdv ¼ 1; rðtÞ ¼ Jvðt;0Þ. (14)

3. Stationary solution and response analysis

When the system is in the stationary states, qpv=qt ¼ 0, dmk=

dt ¼ 0, dgk=dt ¼ 0, pvðt; vÞ ¼ p0v ðvÞ, rðtÞ ¼ r0, mkðtÞ ¼ m0

k , gkðtÞ ¼ g0k

and lkðtÞ ¼ l0k are time-independent. From (9), (12)–(14), we get

p0v ðvÞ ¼

2tvr0

Q0v

exp �ðv� ~K

0

v Þ2

Q0v

" #Z 1

vexp

ðv0 � ~K0

vÞ2

Q0v

" #dv0; 0pvp1,

r0 ¼ tv

ffiffiffiffipp

Z 1� ~K0

v=ffiffiffiffiffiQ0

v

p

� ~K0

v=ffiffiffiffiffiQ0

v

p expðu2Þ erf~K

0

vffiffiffiffiffiffiQ0

v

q0B@

1CAþ erfðuÞ

264375du

0B@1CA�1

,

~K0

v ¼ Se þXN

k¼1

tvAl0km0

k ; Q0v ¼

XN

k¼1

tvA2l0kg

0k ,

m0k ¼

1

1þ Uktdl0k

; g0k ¼

2m0k

2þ tdð2Uk � U2k Þl

0k

. (15)

ARTICLE IN PRESS

W. Huang et al. / Neurocomputing 72 (2008) 630–635632

Sometimes, we are more interested in the instantaneous responseto time-dependence random fluctuation inputs. The inputs takethe form

lk ¼ l0kð1þ ekl

1k ðtÞÞ, (16)

where 0oek51. Then mk and gk have the forms, i.e.,

mk ¼ m0kð1þ ekm1

k ðtÞ þ Oðe2k ÞÞ,

gk ¼ g0kð1þ ekg1

k ðtÞ þ Oðe2k ÞÞ. (17)

~Kv and Qv are

~Kv ¼ Se þXN

k¼1

tvAl0km0

k þXN

k¼1

ektvAl0km0

kðl1k þm1

k Þ þ Oðe2k Þ,

Qv ¼XN

k¼1

tvA2l0kg

0k þ

XN

k¼1

ektvA2l0kg

0k ðl

1k þ g

1k Þ þ Oðe2

k Þ. (18)

Substituting (17) into (12), and ignoring the high order item,it yields

dm1k

dt¼ �

1

tdþ Ukl

0k

� �m1

k � Ukl0kl

1k ðtÞ,

dg1k

dt¼ �

2

tdþ ð2Uk � U2

k Þl0k

� �g1

k þ2m1

k

td� ð2Uk � U2

k Þl0kl

1k ðtÞ. (19)

With the definitions

~Kv ¼~K

0

v þ � ~K1

v ðtÞ þ Oð�2Þ,

Qv ¼ Q0v þ �Q

1v ðtÞ þ Oð�2Þ,

pv ¼ p0v þ �p1ðtÞ þ Oð�2Þ,

r ¼ r0 þ �r1ðtÞ þ Oð�2Þ, (20)

where 0o�51, and boundary conditions of p1

p1ðt;1Þ ¼ 0;

Z 1

0p1ðt; vÞdv ¼ 0, (21)

using the perturbative expansion in powers of �, we can get

0 ¼ �qqvð�vþ ~K

0

vÞp0vðvÞ þ

Qv

2

q2p0vðvÞ

qv2,

tvqp1

qt¼ �

qqvð�vþ ~K

0

vÞp1 þQ0

v

2

q2p1

qv2�qf 0ðt; vÞ

qv,

f 0ðt; vÞ ¼~K

1

v ðtÞp0v �

Q1v ðtÞ

2

qp0v

qv,

r1 ¼ �Q0

v

2tv

qp1ðt;1Þ

qv�

Q1v ðtÞ

2tv

qp0v ð1Þ

qv. (22)

For the oscillatory inputs ~K1

v ðtÞ ¼ kðoÞejot , Q1v ðtÞ ¼ qðoÞejot , the

output has the same frequency and takes the forms

p1ðt; vÞ ¼ poðo; vÞejot , qp1=qt ¼ jop1.For inputs that vary on a slow enough time scale, satisfy

tvo51, we define

�l ¼ tvo,

p1 ¼ p01 þ �lp

11 þ Oð�2

l Þ,

r1 ¼ r01 þ �lr

11 þ Oð�2

l Þ. (23)

Using the perturbative expansion in powers of �l, we get

qf 0ðt; vÞ

qv¼ �

qqvð�vþ ~K

0

vÞp01 þ

Q0v

2

q2p01

qv2,

jp01 ¼ �

qqvð�vþ ~K

0

vÞp11 þ

Q0v

2

q2p11

qv2. (24)

The solutions of Eq. (24) are

pn1 ¼

2

Q0v

exp �ðv� ~K

0

vÞ2

Q0v

" #Z 1

vðtvrn

1 � FnÞ expðv0 � ~K

0

vÞ2

Q0v

" #dv0,

rn1 ¼

2r0

Q0v

Z 1

0exp �

ðv� ~K0

vÞ2

Q0v

" #Z 1

vFn exp

ðv0 � ~K0

vÞ2

Q0v

" #dv0 dv,

F0 ¼ f 0ðt; vÞ; F1 ¼ j

Z v

0p0

1ðv0Þdv0; n ¼ 0;1. (25)

In general, Q1v ðtÞ5

~K1

v ðtÞ, then we have

F0 ¼ f 0ðt; vÞ �~K

1

v ðtÞp0v . (26)

From (23), (25) and (26), we can get

r1 �2r0

Q0v

~K1

v ðtÞ

Z 1

0exp �

ðv� ~K0

vÞ2

Q0v

" #Z 1

vp0

v expðv0 � ~K

0

vÞ2

Q0v

" #dv0 dv

þ jotv2r0

Q0v

Z 1

0exp �

ðv� ~K0

v Þ2

Q0v

" #Z 1

v

Z v0

0p0

1ðv00Þdv00

� �

� expðv0 � ~K

0

vÞ2

Q0v

" #dv0 dv. (27)

In the limit of high frequency inputs, i.e., 1=tvo51, with thedefinitions

�h ¼1

tvo,

p1 ¼ p0h þ �hp1

h þ Oð�2hÞ, (28)

we obtain

p0h ¼ 0; p1

h ¼ jqf 0ðt; vÞ

qv,

r1 ¼ �Q1

v ðtÞ

2tv

qp0vð1Þ

qv� j�h

Q0v

2tv

q2f 0ðt;1Þ

qv2þ Oð�2

hÞ

�Q1

v ðtÞ

Q0r0 � j�h

Q0v

2tv

~K1

v ðtÞq2p0

vð1Þ

qv2�

Q1v ðtÞ

2

q3p0v

qv3

!

¼Q1

v ðtÞr0

Q0v

�2j�h

~K1

v ðtÞr0

Q0v

ð1� ~K0

vÞ �Q1

v ðtÞ

~K1

v ðtÞQ0v

ð1� ~K0

v � Q0v Þ

!.

(29)

When Q1v ðtÞ5

~K1

v ðtÞ, we have

r1 �Q1

v ðtÞr0

Q0v

�2j ~K

1

v ðtÞr0

tvoQ0v

ð1� ~K0

vÞ 1�Q1

v ðtÞ

~K1

v ðtÞQ0v

!, (30)

4. Discussion

In Eq. (15), ~K0

v reflects the average intensity of back-ground inputs and Q0

v reflects the intensity of background noise.

ARTICLE IN PRESS

W. Huang et al. / Neurocomputing 72 (2008) 630–635 633

When 15tdUkl0k , we have

~K0

v � Se þXN

k¼1

tvA

tdUk,

Q0v �

XN

k¼1

tvA2

tdUkð1þ tdUkl0k ð1� Uk=2ÞÞ

. (31)

From (31), we can know the change of background inputs l0k has

little influence on ~K0

v which is dominated by parameter tvA=tdUk,but more influence on Q0

v which decreases with l0k increasing.

In the low input frequency regime, from (27), we can knowthat the input frequency o increasing will result in the responseamplitude and the phase delay increasing. However, in the high

200

100

01.5

1

0.5 00.05

0.1

Q0v

K 0v

15

10

5

Fig. 1. Response amplitude versus Q0v and eK0

v . (A) r1= ~K1

v

��� ��� (for Eq. (27)) cha

16

14

12

10

8

60 0.5 1 1.5 2

0 0.5 1 1.5 2

50

45

40

35

30

1

1

1

1

1

4

4

4

4

3

3

Fig. 2. Simulation of a network of 2000 neurons (thin solid line) and the analytic solution

N ¼ 30, o ¼ 6:28 Hz, l1k ¼ sinðotÞ, ekl

0k ¼ 10 Hz, l0

k ¼ 70 Hz ((A) and (C)) and 100 Hz ((B

(0–2 s), and the longitudinal axis is the fire rate.

input frequency limit regime, from (30), we can know the inputfrequency o increasing will result in the response amplitude andthe phase delay decreasing. Moreover, from (27) and (30), weknow the stationary background fire rate r0 play an importantpart in response to changes in fluctuation outputs. The instanta-neous response r1 increases monotonically with background firerate r0. But the background fire rate r0 is a function of the

background noise Q0v . In Eq. (27), kr1= ~K

1

vk reflects the response

amplitude, and in Eq. (30), r0=Q0v reflects the response amplitude.

As Fig. 1(A) and (B) show that kr1= ~K1

vk and r0=Q0v changes with

variables Q0v and ~K

0

v , respectively. We can know, for the

subthreshold regime ( ~K0

vo1), they increase monotonically with

Q0v

K 0v

000

000

000

01.5

0.51

00.05

0.1

nges with Q0v and ~K

0

v . (B) r0=Q0v (for Eq. (30)) changes with Q0

v and ~K0

v .

0 0.5 1 1.5 2

0 0.5 1 1.5 2

4

3

2

1

0

9

8

7

6

4

2

0

8

6

(thick solid line) for Eqs. (15) and (27), with tv ¼ 15 ms, td ¼ 1 s, A ¼ 0:5, Uk ¼ 0:5,

) and (D)), Se ¼ 0:5 ((A) and (B)) and 0.8 ((C) and (D)). The horizontal axis is time

ARTICLE IN PRESS

W. Huang et al. / Neurocomputing 72 (2008) 630–635634

Q0v when ~K

0

v is a constant. However, for the suprathreshold regime

( ~K0

v41), they decrease monotonically with Q0v when ~K

0

v is a

constant.When inputs remain, if the instantaneous response amplitude

increases, then we can take for the role of neurons are more likecoincidence detection than temporal integration. And from thisviewpoint, it suggests that the background inputs play animportant role in information processing and act as a switchbetween temporal integration and coincidence detection.

In Eq. (16), if the inputs take the oscillatory form, l1k ðtÞ ¼ ejot ,

according to (19), we get

m1k ¼ �

tdUkl0kejðot�ymÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðtdoÞ2 þ ð1þ tdUkl0kÞ

2q , (32)

where ym ¼ arctgðtdo=ð1þ tdUkl0kÞÞ is the phase delay, tdUkl

0k=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðtdoÞ2 þ ð1þ tdUkl0k Þ

2q

is the amplitude. The minus shows it is a

‘depression’ response amplitude. The phase delay increases withthe input frequency o and decreases with the background input

l0k . The ‘depression’ response amplitude decreases with the input

frequency o and increases with the background input l0k .

Eqs. (15), (18), (12), (19), (27), (30) and (32) show us a point ofview that the synapses can be regarded as a time-dependentexternal field which impacts on the neuronal population throughthe time-dependent mean and variance. We assume the inputs are

composed of two parts, viz. l1k1ðtÞ ¼ l1

k2ðtÞ ¼ 1

2ejot , then we can get

m1k1

and m1k2

. However, in general m1kam1

k1þm1

k2, this suggest for

us that the spatial distribution of synapses and inputs isimportant on neural information processing. In conclusion, therole of synapses can be regarded as a spatio-temporalfilter. Fig. 2is the results of simulation of a network of 2000 neurons and theanalytic solution for Eqs. (15) and (27) in different conditions.

5. Summary

In this paper, we deal with the model of the integrate-and-fire(IF) neurons with synaptic current dynamics and synapticdepression. In Section 2, first, using the membrane potentialequation (1) and combining the synaptic depression equation (2),we derive the evolution equation (4) of the joint distributiondensity function. Then, we give an approach to cut the evolutionequation of the high-dimensional function down to one dimen-sion, and get Eq. (9). Finally, we give the stationary solutionand the response of instantaneous fire rate to time-dependencerandom fluctuation inputs. In Section 3, the analysis and discussionof the model is given and several significant conclusions arepresented. This paper can only investigate the IF neuronal modelwithout internal connection. We can also extend to other models,such as the non-linear IF neuronal models of sparsely connectednetworks of excitatory and inhibitory neurons.

Acknowledgments

The authors wishes to thank the anonymous reviewersfor their valuable comments. This work was supported bythe National Natural Science Foundation of China (Grant nos.60703107, 60703108), the National High Technology Researchand Development Program (863 Program) of China (Grant no.2006AA01Z107), the National Basic Research Program (973Program) of China (Grant no. 2006CB705700) and the Programfor Cheung Kong Scholars and Innovative Research Team inUniversity (PCSIRT, IRT0645).

Appendix

We consider the following equation:

dxðtÞ

dt¼ cðxÞ þ gðxÞdðt � tspÞ, (33)

tsp obeys a Poisson distribution with mean s. We have transitionprobability

Wðz; t þ Dtjx; tÞ ¼ ð1� DtsÞdðz� ðxþ cðxÞDtÞÞ

þ Dtsdðz� ðxþ cðxÞDt þ gðxÞÞÞ. (34)

Our aim is to investigate the density evolution equation of xðtÞ,pðz; tÞ. According to (34), we have

pðz; t þDtÞ ¼

ZWðz; t þDtjx; tÞpðx; tÞdx

¼

Zdðz� ðxþ cðxÞDtÞÞpðx; tÞdx

þ DtsZ dðz� ðxþ cðxÞDt þ gðxÞÞÞ

�dðz� ðxþ cðxÞDtÞÞ

!pðx; tÞdx (35)

and

qpðz; tÞ

qt¼ lim

Dt!0

pðz; t þ DtÞ � pðz; tÞ

Dt¼ �

qqzðcðzÞpðz; tÞÞ

þ sZðdðz� ðxþ gðxÞÞÞ � dðz� xÞÞpðx; tÞdx, (36)

where

dðz� ðxþ gðxÞÞÞ ¼ dðz� xÞ � gðxÞd0ðz� xÞ þg2ðxÞ

2d00ðz� xÞ þ Oðg3ðxÞÞ,

(37)

d0ð�Þ is the derivative of dð�Þ. If Oðg3ðxÞÞ ! 0, then we can ignoreOðg3ðxÞÞ, and substituting (37) into (36), obtain

qpðz; tÞ

qt¼ �

qqzðcðzÞpðz; tÞÞ þ s

Z�gðxÞd0ðz� xÞ þ

g2ðxÞ

2d00ðz� xÞ

� ��pðx; tÞd

¼ �qqzððcðzÞ þ sgðzÞÞpðz; tÞÞ þ

s2

q2

qz2ðg2ðzÞpðz; tÞÞ. (38)

We can generalize the above method to the condition that ismultidimensional.

References

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Wentao Huang was born in Hubei, China, on Novem-ber 29, 1978. He received the B.S. and M.S. degrees insignal processing from Xidian University, Xi’an, China,in 2001 and 2004, respectively, and is currentlypursuing the Ph.D. degree in pattern recognition andintelligent information system from the Institute ofIntelligent Information Processing, Xidian University,Xi’an, China. His research interests include computa-tional neuroscience and computational vision.

Licheng Jiao (SM’89) was born in Shaanxi, China, onOctober 15, 1959. He received the B.S. degree fromShanghai Jiaotong University, Shanghai, China, in 1982,and the M.S. and Ph.D. degrees from Xi’an JiaotongUniversity, Xi’an, China, in 1984 and 1990, respectively.From 1984 to 1986, he was an Assistant Professor inCivil Aviation Institute of China, Tianjing, China. During1990 and 1991, he was a Postdoctoral Fellow in theNational Key Lab for Radar Signal Processing, XidianUniversity, Xi’an, China. Since 1992, he has been withthe National Key Lab for Radar Signal Processing,where he became a full Professor. Now he is also the

Dean of the Electronic Engineering School and the

Institute of Intelligent Information Processing at Xidian University. His currentresearch interests include signal and image processing, nonlinear circuit andsystems theory, learning theory and algorithms, computational vision, computa-tional neuroscience, optimization problems, wavelet theory, and data mining. He is

the author of there books: Theory of Neural Network Systems (Xi’an, China: XidianUniversity Press, 1990), Theory and Application on Nonlinear TransformationFunctions (Xi’an, China: Xidian University Press, 1992), and Applications andImplementations of Neural Networks (Xi’an, China: Xidian University Press, 1996).He is the author or coauthor of more than 150 scientific papers.

Jianhua Jia was born in Jiangxi, China, on September25, 1979. He received the B.S. and M.S. degrees in signalprocessing from Nanchang University and XidianUniversity, China, in 2000 and 2006, respectively, andis currently pursuing the Ph.D. degree in patternrecognition and intelligent information system fromthe Institute of Intelligent Information Processing,Xidian University, Xi’an, China. His research interestsinclude image processing and computational neuro-science.