Lecture Notes in Computer Science

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4 Conclusion
Acknowledgment

3.2 C-Shape

We tested the model with a C-shaped figure that has been suggested to be difficult

shape for the determination of BO. Fig. 4 shows the simulation result for the C-shape.

The responses of a horizontal and vertical cross sections, as indicated in Fig.3(A), are

shown here in Fig. 4(B) and (C), respectively, with the conventions same as Fig.1.

The figure exhibits the firing rates for two types of the model cells, BO model cells

responding to contours (solid lines) and MA model cells (dotted lines). MA represen-

tation predicts a strong peak at 0 along the horizontal cross section, and distributed

strong responses along the vertical cross section. Although we observe the maximum

responses at the centers corresponding to MA, the peak along the horizontal was not

significant, and the distribution along the vertical was peaky and uneven. It appears

that MA cells cannot integrate properly the signals propagated from BO cells. This

distributed MA response comes from complicated propagations of BO signals from

the concaved shape. Furthermore, as has been suggested [3], the C-shaped figure is a

challenging shape for the determination of BO. Therefore, the BO signals are not

clear before the propagation begins. We would like to note that the model is still ca-

pable of providing a basis for MA representation for a complicated figure.

(A)

-1 0 1

Horizontal position

ike

-1 0 1

Vertical position

ike

(B) (C)

Fig. 4.

The simulation results for the C-shaped figure. (A) Positions of the analysis along hori-

zontal and vertical cross-sections as indicated by dashed line. The responses of the model cells

in firing rate along the horizontal cross section (B), and that along the vertical cross section (C).

Solid and dotted lines indicate the responses of BO and MA model cells, respectively. Although

the maximum responses are observed at the centers, the responses are distributed.

3.3 Natural Images

The model has shown its ability to extract a basis for MA representation for not only

simple but also difficult shapes. To further examine the model for arbitrary shapes, we

Representation of Medial Axis from Synchronous Firing of BO Selective Cells

tested the model with natural images taken from Berkeley Segmentation Dataset [8].

Fig.2(C) shows an example, an eagle stops on a tree branch. Because we are inter-

ested in the representation of shape, we extracted its shape by binarizing the gray

scale, as shown in Fig. 5(A). The simulation results of BO and MA model cells are

shown in Fig. 5(B). We plotted the responses along the horizontal cross-section indi-

cated in Fig. 5(A). Although the shape is much more complicated, detailed and

asymmetric, the results are very similar to that for a square as shown in Fig.2(B). The

BO model cells responded to the contours (horizontal positions at -1 and 1), and MA

model cells exhibited a strong peak at the center (horizontal position at 0). This result

indicates that the model detects MA for figures with arbitrary shape. The aim of the

simulations with natural images is to test a variety of stimulus shape and configura-

tions that are possible in natural scenes. Further simulations with a number of natural

images is expected, specifically with images including occulusion, multiple objects

and ambiguous figures.

-1 0 1

Horizontal position

ik

es

(A) (B)

Fig. 5. An example of simulation results for natural images. (A) The binary image of an eagle

together with the horizontal cross-section for the graphical presentation of the results. (B) The

responses of the model cells in firing rate along the cross-section as a function of the horizontal

location. Horizontal positions -1 and 1 represent the places on the vertical edges of the eagle.

Zero represents the center of the bird. A clear peak at the center corresponding to MA is ob-

served. Although the shape is much more complicated, detailed and asymmetric, the results are

very similar to that for a square as shown in Fig. 2(B), indicating robustness of the model.

4 Conclusion

We studied whether early visual areas could provide basis for MA representation,

specifically what constraint is necessary for the representation of MA. Our results

showed that simultaneous firing of BO-selective neurons is crucial for MA represen-

tation. We implemented the physiologically realistic firing model neurons that have

connections from BO model cells to MA model cells. If a stimulus is presented at

once so that BO cells along the contours fire simultaneously, and if a MA cell is lo-

cated equidistant from some of contours of the stimulus, then the MA cell fires be-

cause of synchronous signals from the BO cells that give rise to strong EPSP. We

showed three typical examples of the simulation results, a simple square, a difficult

C-shaped figure, and a natural image of an eagle. The simulation results showed that

the model provides a basis for MA representation for all three types of stimuli. These

Y. Hatori and K. Sakai

results suggest that the simultaneous firing of BO cells is an essence for the MA rep-

resentation in early visual areas.

Acknowledgment

We thank Dr. Haruka Nishimura for her insightful comments and Mr. Satoshi Wata-

nabe for his help in simulations. This work was supported by Grant-in-aid for Scien-

tific Research from the Brain Science Foundation, the Okawa Foundation, JSPS

(19530648), and MEXT of Japan (19024011).

References

1. Zhou, H., Friedman, H.S., Heydt, R.: Coding of Border Ownership in Monkey Visual Cor-

tex. The Journal of Neuroscience 86, 2796–2808 (2000)

2. Lee, T.S., Mumford, D., Romero, R., Lamme, V.A.F.: The role of the primary visual cortex

in higher level vision. Vision Research 38, 2429–2454 (1998)

3. Sakai, K., Nishimura, H.: Surrounding Suppression and Facilitation in the determination of

Border Ownership. The Journal of Cognitive Neuroscience 18, 562–579 (2006)

4. NEURON: http://www.neuron.yale.edu/neuron/

5. Johnston, D., Wu., S. (eds.): Foundations of Cellular Neurophysiology. MIT Press, Cam-

bridge (1999)

6. Carandini, M., Heeger, D.J., Movshon, J.A.: Linearity and Normalization in Simple Cells of

the Macaque Primary Visual Cortex. The Journal of Neuroscience 21, 8621–8644 (1997)

7. Jones, H.E., Wang, W., Silito, A.M.: Spatial Organization and Magnitude of Orientation

Contrast Interaction in Primate V1. Journal of Neurophysiology 88, 2796–2808 (2002)

8. The Berkeley Segmentation Dataset: http://www.eecs.berkeley.edu/Research/

Projects/CS/vision/grouping/segbench/

9. Engel, A.K., Fries, P., Singer, W.: Dynamic predictions: oscillations and synchrony in top-

down processing. Nature Reviews Neuroscience 2, 704–716 (2001)

Neural Mechanism for Extracting Object

Features Critical for Visual Categorization Task

Mitsuya Soga

and Yoshiki Kashimori

1,2

Dept. of Information Network Science, Graduate school of Information Systems,

Univ. of Electro-Communications, Chofu, Tokyo 182-8585, Japan

Dept. of Applied Physics and Chemistry, Univ. of Electro-communications, Chofu,

Tokyo 182-8585, Japan

Abstract. The ability to group visual stimuli into meaningful cate-

gories is a fundamental cognitive process. Some experiments are made to

investigate the neural mechanism of visual categorization. Although ex-

perimental evidence is known that prefrontal cortex (PFC) and infe-

riortemporal (IT) cortex neurons sensitively respond in categorization

task, little is known about the functional role of interaction between

PFC and IT in categorization task To address this issue, we propose a

functional model of visual system, and investigate the neural mechanism

for the categorization task of line drawings of faces. We show here that

IT represents similarity of face images based on the information of the

resolution maps of early visual stages. We show also that PFC neurons

bind the information of part and location of the face image, and then

PFC generates a working memory state, in which only the information

of face features relevant to the categorization task are sustained.

Introduction

Visual categorization is fundamental to the behavior of higher primates. Our raw

perceptions would be useless without our classiﬁcation of items such as animals

and food. The visual system has the ability to categorize visual stimuli, which is

the ability to react similarity to stimuli even when they are physically distinct,

and to react diﬀerently to stimuli that may be similar. How does the brain group

stimuli into meaningful categories?

Some experiments have been made to investigate the neural mechanism of

visual categorization. Freedman et al.[1] examined the responses of neurons in the

prefrontal cortex(PFC) of monkey trained to categorize animal forms(generated

by computer) as either “doglike” or “catlike”. They reported that many PFC

neurons responded selectively to the diﬀerent types of visual stimuli belonging

to either the cats or the dogs category. Sigala and Logothetis [2] recorded from

inferior temporal (IT) cortex after monkey learned a categorization task, and

found that selectivity of the IT neurons was signiﬁcantly increased to features

critical for the task. The numerous reciprocal connections between PFC and IT

could allow the necessary interactions to select the best diagnostic features of

stimuli [3]. However, little is known about the role of interaction between IT

M. Ishikawa et al. (Eds.): ICONIP 2007, Part I, LNCS 4984, pp. 27–36, 2008.

c Springer-Verlag Berlin Heidelberg 2008

M. Soga and Y. Kashimori

and PFC in categorization task that gives the category boundaries in relation

to behavioral consequences.

To address this issue, we propose a functional model of visual system in which

categorization task is achieved based on functional roles of IT and PFC. The

functional role of IT is to represent features of object parts, based on diﬀerent

resolution maps in early visual system such as V1 and V4. In IT, visual stimuli

are categorized by the similarity based on the features of object parts. Posterior

parietal (PP) encodes the location of object part to which attention is paid.

The PFC neurons combine the information about feature and location of object

parts, and generate a working memory of the object information relevant to the

categorization task. The synaptic connections between IT and PFC are learned

so as to achieve the categorization task. The feedback signals from PFC to IT

enhance the sensitivity of IT neurons that respond to the features of object parts

critical for the categorization task, thereby enabling the visual system to perform

quickly and reliably task-dependent categorization.

In the present study, we present a neural network model which makes cate-

gories of visual objects depending on categorization task. We investigated the

neural mechanism of the categorization task about line drawings of faces used by

Sigala and Logothetis [2]. Using this model we show that IT represents similarity

of face images based on the information of the resolution maps in V1 and V4.

We also show that PFC generates a working memory state, in which only the

information of face features relevant to the categorization task are sustained.

Model

To investigate the neural mechanism of visual categorization, we made a neural

network model for a form perception pathway from retina to prefrontal cor-

tex(PFC). The model consists of eight neural networks corresponding to the

retina, lateral geniculate nucleus(LGN), V1, V4, inferior temporal cortex (IT),

posterior parietal(PP), PFC, and premotor area, which are involved in ventral

and dorsal pathway [4,5]. The network structure of our model is illustrated in

Fig. 1.

2.1

Model of Retina and LGN

The retinal network is an input layer, on which the object image is projected.

The retina has a two-dimensional lattice structure that contains N

x N

pixel

scene. The LGN network consists of three diﬀerent types of neurons with re-

spect to the spatial resolution of contrast detection, ﬁne-tuned neurons with

high spatial frequency (LGNF), middle-tuned neurons with middle spatial fre-

quency (LGNM), and broad-tuned neurons with low spatial frequency (LGNB).

The output of LGNX neuron (X=B, M, F) of (i, j) site is given by

LGN

, j

; x) =

, j

)M (i, j; x),

(1)

Neural Mechanism for Extracting Object Features

Fig. 1. The structure of our model. The model is composed of eight modules structured

such that they resemble ventral and dorsal pathway of the visual cortex, retina, lateral

geniculate nucleus (LGN), primary visual cortex (V1), V4, inferior temporal cortex

(IT), posterior parietal (PP), prefrontal cortex (PFC), and premotor area. a1

∼ a3

mean dynamical attractors of visual working memory.

M (i, j; x) = A exp

−

(i

− i

)

+ (j

− j

)

−B exp −

− i

)

+ (j

− j

)

(2)

where, I

, j

) is the gray-scale intensity of the pixel of retinal site (i

, j

and the function M (i, j; X) is Mexican hat-like function that represents the

convergence of the retinal inputs with ON center-OFF surrounding connections

between retina and LGN. The parameter values were set to be A = 1.0, B = 1.0,

1x

= 1.0, and σ

= 2.0.

2.2

Model of V1 Network

The neurons of V1 network have the ability to elemental features of the object

image, such as orientation and edge of a bar. The V1 network consists of three

diﬀerent types of networks with high, middle, and broad spatial resolutions, V1B,

V1M, and V1F, each of which receives the outputs of LGNB, LGNM, and LGNF,

respectively. The V1X network (X=B, M, F) contains M

× M

hypercolumns,

each of which contains L

orientation columns. The neurons in V1X (X=B,

M, F) have receptive ﬁeld performing a Gabor transform. The output of V1X

neuron of (i, j) site is given by

V 1

(i, j, θ; x) =

LGN

(p, q; x)G(p, q, θ; x),

(3)

M. Soga and Y. Kashimori

G(p, q, θ; x) =

2πσ

exp

−

− p)

− q)

× sin 2πf

p cos θ + 2πf

q sin θ +

(4)

where f

, f

are spatial frequencies of x- and y- coordinate, respectively. The

parameter values were σ

= 1, σ

= 1, f

= 0.1Hz, and f

= 0.1Hz.

2.3

Model of V4 Network

The V4 network consists of three diﬀerent networks with high, middle, and

low spatial resolutions, which receive convergent inputs from the cell assemblies

with the same tuning in V1F, V1M, and V1B. The convergence of outputs of V1

neurons enables V4 neurons to respond speciﬁcally to a combination of elemental

features such as a cross and triangle represented on the V4 network.

2.4

Model of PP

The posterior parietal(PP) network consists of N

P P

x N

P P

neurons, each of

which corresponds to a spatial position of each pixel of the retinal image. The

functions of PP network are to represent the spatial position of a whole object

and the spatial arrangement of its parts in the retinotopic coordinate system

and to mediate the location of the object part to which attention is paid.

2.5

Model of IT

The network of IT consists of three subnetworks, each of which receives the

outputs of V4F, V4M, and V4B maps, respectively. Each network has neurons

tuned to various features of the object parts depending on the resolutions of V4

maps and the location of the object parts to which attention is directed. The ﬁrst

sub-network detects a broad outline of a whole object, the second subnetwork

detects elemental ﬁgures of the object that represent elemental outlines of the

object, and the third subnetwork represents the information of the object parts

based on the ﬁne resolution of V4F map. Each subnetwork was made based on

Kohonen’s self-organized map model. The elemental ﬁgures in the second sub-

network may play an important role in extracting the similarity to the outlines

of objects.

2.6

Model for Working Memory in PFC

The PFC memorizes the information of spatial positions of the object parts as

dynamical attractors. The functional role of the PFC is to reproduce a complete

form of the object by binding the information of the object parts memorized

in the second and third subnetworks of ITC and the their spatial arrangements

represented by PP network. The PFC network model was made based on the

dynamical map model [6,7]. The network model consists of three types of neu-

rons, M, R, and Q neurons. Q neuron is connected to M neuron with inhibitory

Neural Mechanism for Extracting Object Features

synapse. M neurons are interconnected to each other with excitatory and in-

hibitory synaptic connections. M neuron layer of the present model corresponds

to an associative neural network. M neurons receive inputs from three IT sub-

networks.

Dynamical evolutions of membrane potentials of the neurons, M, R, and Q,

are described by

−u

mm,ij

(t, T

− τ

) + W

U qi

IT,ik

P P,il

P P

(5)

−u

+ W

(6)

−u

+ W

(7)

where u

m,i

, u

q,i

, and u

r,i

are the membrane potentials of ith M neuron, ith Q

neuron, and ith R neuron, respectively. τ

, τ

are the relaxation times of

these membrane potentials. τ

is the delay time of the signal propagation from

jth M neuron to ith one, and τ

max

is the maximum delay time. The time delay

plays an important role in the stabilization of temporal sequence of ﬁring pattern.

mm,ij

(t, T

) is the strength of the axo-dendric synaptic connection from jth M

neuron to ith M neuron whose propagation delay time is τ

. W

(t), W

(t),

and W

(t), and W

(t) are the strength of the dendro-dendritic synaptic con-

nection from Q neuron to M neuron, from R neuron to M neuron, from M neuron

to Q neuron, and from M neuron to R neuron, respectively. V

is the output of

ith M neuron, U

and U

are dendritic outputs of ith Q neuron and ith R neu-

ron, respectively. Outputs of Q and R neurons are given by sigmoidal functions

of u

and u

, respectively. M neuron has a spike output, the ﬁring probability

of which is determined by a sigmoidal function of u

. W

IT,ik

and W

P P,il

are

the synaptic strength from kth IT neuron to ith M neuron and that from lth

PP neuron to ith M neuron, respectively. X

k

and X

P P

are the output of k

th IT neuron and lth PP neuron, respectively. The parameters were set to be

= 3ms, τ

= 1ms, τ

= 1ms, W

= 1, W

−10,

and τ

= 38ms.

Dynamical evolution of the synaptic connections are described by

mm,ij

(t, T

)

−W

mm,ij

(t, T

) + λV

(t)V

− T

(8)

where τ

is a time constant, and λ is a learning rate.The parameter values were

= 1200ms, and λ = 28. The PFC connects reciprocally with IT, PP, and the

premotor networks. The synaptic connections between PFC and IT and those

between PFC and PP are learned by Hebbian learning rule.

M. Soga and Y. Kashimori

2.7

Model of Premotor Cortex

The model of premotor area consists of neurons whose ﬁring correspond to action

relevant to the categorization task, that is, pressing right or left lever.The details

of mathematical description are described in Refs. [8-10].

Neural Mechanism for Extracting Diagnostic Features

in Visual Categorization Task

3.1

Categorization Task

We used the line drawings of faces used by Sigala and Logothetis [2] to inves-

tigate the neural mechanism of categorization task. The face images consist of

four varying features, eye height, eye separation, nose length, and mouse height.

The monkeys were trained to categorize the face stimuli depending on two di-

agnostic features, eye height and eye separation. The two diagnostic features

allowed separation between classes along a linear category boundary as shown

in Fig.2b. The face stimuli were not linearly separable by using the other two,

non-diagnostic features, or nose length and mouth height. On each trial, the

monkeys saw one face stimulus and then pressed one of two levers to indicate

category. Thereafter, they received a reward only if they chose correct category.

After the training, the monkeys were able to categorize various face stimuli based

on the two diagnostic features. In our simulation, we used four training stimuli

shown in Fig.2a and test stimuli with varying four features.

Fig. 2. a) The training stimulus set consisted of line drawing of faces with four varying

features: eye separation, eye height, nose length and mouth height. b) In the categoriza-

tion task, the monkyes were presented with one stimulus at a time. The two categories

were linearly separable along the line. The test stimuli are illustrated by the marks, ‘x‘

and ‘o‘. See Ref.2 for details of the task.

Neural Mechanism for Extracting Object Features

3.2

Neural Mechanism for Accomplishing Visual Categorization

Task

Neural mechanism for accomplishing visual categorization task is illustrated in

Fig.3. Object features are encoded by hierarchical processing at each stage of ven-

tral visual pathway from retina to V4. The IT neurons encode the information of

object parts such as eyes, nose, and mouth. The PP neurons encode the location

of object parts to which attention should be directed. The information of object

part and its location are combined by a dynamical attractor in PFC. The output

of PFC is sent to two types of premotor neurons, each ﬁring of which leads to

pressing of the right or left lever. When monkeys exhibit the relevant behavior for

the task and then receive a reward, the attractor in PFC, which represents the

object information relevant to the task, is gradually stabilized by the facilitation

of learning across the PFC neurons and that between PFC and premotor neurons.

On the other hand, when monkeys exhibit the irrelevant behavior for the task

and then receive no reward, the attractor associated with the irrelevant behavior

is destabilized, and then eliminated in the PFC network. As a result, the PFC re-

tains only the object information relevant to the categorization task, as working

memory. The feedback from PFC to IT and PP makes the responses of IT and

PP neurons strengthened, thereby enabling the visual system to rapidly and ac-

curately discriminate between object features belonging to diﬀerent categories.

When monkey pays attention to a local region of face stimulus, the attention

signal from other brain area such as prefrontal eye ﬁeld increases the activity of

PP neurons encoding the location of face part to which attention is directed. The

PP neurons send their outputs back to V4, and thereby increasing the activity

of V4 neurons encoding the feature of the face parts which the attention is paid,

because V4 has the same retinotopic map as PP. Thus the attention to PP allows

V4 to send IT only the information of face part to which attention is directed,

Fig. 3. Neural mechanism for accomplishing visual categorization task. The informa-

tion about object part and it’s location are combined to generate working memory,

indicated by α and β. The solid and dashed lines indicate the formation and elimina-

tion of synaptic connection, respectively.

M. Soga and Y. Kashimori

leading to generation of IT neurons encoding face parts. Furthermore, as the

training proceeds, the attention relevant to the task is ﬁxed by the learning of

synaptic connections between PFC and PP, allowing monkey to perform quickly

the visual task.

Results

4.1

Information Processing of Visual Images in Early Visual Areas

Figure 4 shows the responses of neurons in early visual areas, LGN, V1, and

V4, to the training stimulus, face 1 shown in Fig. 2a. The visual information is

processed in a hierarchical manner, because the neurons involved in the path-

way from LGN to V4 have progressively larger receptive ﬁelds and prefer more

complex stimuli. At the ﬁrst stage, the contrast of the stimulus is encoded

by ON center-OFF surrounding receptive ﬁeld of LGN neurons, as shown in

Fig. 4b. Then, the V1 neurons, receiving the outputs of LGN neurons, encode the

information of directional features of short bars contained in the drawing face,

as shown in Fig. 4c. The V4 network was made by Kohonen’s self-organized map

so that the V4 neurons could respond to more complex features of the stimulus.

Figure 4d shows that the V4 neurons respond to the more complex features such

as eyes, nose, and mouth.

Fig. 4. Responses of neurons in LGN, V1, and V4. The magnitude of neuronal responses

in these areas is illustrated with a gray scale, in which the response magnitude is

increased with increase of gray color. a) Face stimulus. The image is 90 x 90 pixel

scale. b) Response of LGN neurons. c) Responses of V1 neurons tuned to four kinds of

directions. d) Responses of V4 neurons. The kth neurons (k=1-3) encode the stimulus

features such as eyes, nose, and mouth, respectively.

4.2

Information Processing of Visual Images in IT Cortex

Figure 5a shows the ability of IT neurons encoding eye separation and eye height

to categorize test stimuli of faces. The test stimuli with varying the two features

were categorized by the four ITC neurons learned by the four training stimuli

shown in Fig.2a, suggesting that the ITC neurons are capable for separating test

stimuli into some categories, based on similarity to the features of face parts.

Neural Mechanism for Extracting Object Features

Fig. 5. a) Ability of IT neurons to categorize face stimuli for two diagnostic features.

The four IT neurons were made by using the four kinds of training stimuli shown

in Fig. 2a, whose features are represented by four kinds of symbols (circle, square,

triangle, cross). The test stimuli, represented by small symbols, are categorized by

the four IT neurons. The kind of small symbols means the symbol of IT neuron that

categorizes the test stimulus. The solid lines mean the boundary lines of the categories.

b) Temporal variation of dynamic state of the PFC network during the categorization

task. The attractors representing the diagnostic features are denoted by α

∼ δ, and

the attractors representing non-diagnostic feature is denoted by . A mark on the row

corresponding to α

∼ indicates that the network activity stays in the attractor. The

visual stimulus of face 1 was applied to the retina at 300m

∼ 500ms.

Similarly, the IT neurons encoding nose length and mouth height separated

test stimuli into other categories on the basis of similarity of the two features.

However, the classiﬁcation in the IT is not task-dependent, but is made based

on the similarity of face features.

4.3

Mechanism for Generating Working Memory Attractor in PFC

The PFC combines the information of face features and that of location of face

parts to which attention is directed, and then makes memory attractors about

the information. Figure 5b shows temporal variation of the memory attractors

in PFC. The information about face parts with the two diagnostic features is

represented by attractors X (X= α, β, γ, δ), in which X represents the informa-

tion about eye separation and eye height of the four training stimuli and the

location around eyes. The attractors X are dynamically linked in the PFC. As

shown in Fig. 5b, the information about face parts with the diagnostic features

are memorized as working memory α

∼ δ, because the synaptic connections be-

tween PFC and premotor area are strengthened by a reward signal given by the

choice of correct categorization. On the other hand, the information about face

parts with non-diagnostic features are not memorized as a stable attractor, as

shown by

in Fig.5b, because the information of non-diagnostic features does

M. Soga and Y. Kashimori

not lead to correct categorization behavior. Thus, the PFC can retain only the

information required for the categorization task, as working memory.

Concluding Remarks

In the present study, we have shown that IT represents similarity of face images

based on the resolution maps of V1 and V4, and PFC generates a working

memory state, in which the information of face features relevant to categorization

task are sustained.

The feedback from PFC to IT and PP may play an important role in extracting

the diagnostic features critical for the categorization task. The feedback from

PFC increases the sensitivity of IT and PP neurons which encode the relevant

object feature and location to the task, respectively. This allows the visual system

to rapidly and accurately perform the categorization task. It remains to see how

the feedback from PFC to IT and PP makes the functional connections across

the three visual areas.

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primate temporal cortex. Nature 415, 318–320 (2002)

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An Integrated Neuro-mechanical Model of

C. elegans Forward Locomotion

Jordan H. Boyle, John Bryden, and Netta Cohen

School of Computing, University of Leeds, Leeds LS2 9JT, United Kingdom

Abstract. One of the most tractable organisms for the study of nervous

systems is the nematode Caenorhabditis elegans, whose locomotion in

particular has been the subject of a number of models. In this paper we

present a ﬁrst integrated neuro-mechanical model of forward locomotion.

We ﬁnd that a previous neural model is robust to the addition of a

body with mechanical properties, and that the integrated model produces

oscillations with a more realistic frequency and waveform than the neural

model alone. We conclude that the body and environment are likely to

be important components of the worm’s locomotion subsystem.

Introduction

The ultimate aim of neuroscience is to unravel and completely understand the

links between animal behaviour, its neural control and the underlying molecular

and genetic computation at the cellular and sub-cellular levels. This daunting

challenge sets a distant goal post in the study of the vast majority of animals,

but work on one animal in particular, the nematode Caenorhabditis elegans,

is leading the way. This tiny worm has only 302 neurons and yet is capable of

generating an impressive wealth of sensory-motor behaviours. With the ﬁrst fully

sequenced animal genome [1], a nearly complete wiring diagram of the nervous

circuit [2], and hundreds of well characterised mutant strains, the link between

genetics and behaviour never seemed more tractable.

To date, a number of models have been constructed of subcircuits within the

C. elegans nervous system, including sensory circuits for thermotaxis and chemo-

taxis [3,4], reﬂex control such as tap withdrawal [5], reversals (from forward to

backward motion and vice versa) [6] and head swing motion [7]. Locomotion,

like the overwhelming majority of known motor activity in animals, relies on

the rhythmic contraction of muscles, which are controlled or regulated by neural

networks. This system consists of a circuit in the head (generally postulated to

initiate motion and determine direction) and an additional subcircuit along the

ventral cord (responsible for propagating and sustaining undulations, and po-

tentially generating them as well). Models of C. elegans locomotion have tended

to focus on forward locomotion, and in particular, on the ability of the worm to

generate and propagate undulations down its length [8,9,10,11,12]. These mod-

els have tended to study either the mechanics of locomotion [8] or the forward

locomotion neural circuit [9,10,11,12]. In this paper we present simulations of an

M. Ishikawa et al. (Eds.): ICONIP 2007, Part I, LNCS 4984, pp. 37–47, 2008.

c Springer-Verlag Berlin Heidelberg 2008

J.H. Boyle, J. Bryden, and N. Cohen

integrated model of the neural control of forward locomotion [12] with a min-

imal model of muscle actuation and a mechanical model of a body, embedded

in a minimal environment. The main questions we address are (i) whether the

disembodied neural model is robust to the addition of a body with mechanical

properties; and (ii) how the addition of mechanical properties alters the output

from the motor neurons. In particular, models of the isolated neural circuit for

locomotion suﬀer from a common limitation: the inability to reproduce undula-

tions with frequencies that match the observed behaviour of the crawling worm.

To address this question, we have limited our integrated model to a short section

of the worm, rather than modelling the entire body. We ﬁnd that the addition

of a mechanical framework to the neural control model of Ref. [12] leads to ro-

bust oscillations, with signiﬁcantly smoother waveforms and reduced oscillation

frequencies, matching observations of the worm.

Background

2.1

C . elegans Locomotion

Forwards locomotion is achieved by propagating sinusoidal undulations along

the body from head to tail. When moving on a ﬁrm substrate (e.g. agarose) the

worm lies on its side, with the ventral and dorsal muscles at any longitudinal

level contracting in anti-phase. With the exception of the head and neck, the

worm is only capable of bending in the dorso-ventral plane.

Like all nematode worms, C. elegans lacks any form of rigid skeleton. Its

roughly cylindrical body has a diameter of

∼ 80 μm and a length of ∼ 1 mm. It

has an elastic cuticle containing (along with its intestine and gonad) pressurised

ﬂuid, which maintains the body shape while remaining ﬂexible. This structure

is referred to as a hydrostatic skeleton. The body wall muscles responsible for

locomotion are anchored to the inside of the cuticle.

2.2

The Neural Model

The neural model used here is based on the work of Bryden and Cohen [11,12,13].

Speciﬁcally, we use the model (equations and parameters) presented in [12] which

is itself an extension of Refs. [11,13]. The model simpliﬁes the neuronal wiring

diagram of the worm [2,14] into a minimal neural circuit for forward locomotion.

This reduced model contains a set of repeating units, (one “tail” and ten “body”

units) where each unit consists of one dorsal motor neuron (of class DB) and one

ventral motor neuron (of class VB). A single command interneuron (representing

a pair of interneurons of class AVB in the biological worm) provides the “on”

signal to the forward locomotion circuit and is electrically coupled (via gap

junctions) to all motor neurons of classes DB and VB. In the model, motor

neurons also have sensory function, integrating inputs from stretch-receptors, or

mechano-sensitive ion channels, that encode each unit’s bending angle. Motor

neurons receive both local and – with the exception of the tail – proximate

sensory input, with proximate input received from the adjacent posterior unit.

An Integrated Neuro-mechanical Model of C. elegans Forward Locomotion

Fig. 1. A: Schematic diagram of the physical model illustrating nomenclature (see

Appendix B for details). B: The neural model, with only two units (one body, one tail).

AVB is electrically coupled to each of the motor neurons via gap junctions (resistor

symbols).

The sensory-motor loop for each unit gives rise to local oscillations which phase

lock with adjacent units. Equations and parameters for the neural model are set

out in Appendix A.

This neural-only model uses a minimal physical framework to translate neu-

ronal output to bending. Fig. 1B shows the neural model with only two units (a

tail and one body unit), as modelled in this paper. In the following section, we

outline a more realistic physical model of the body of the worm.

Physical Model

Our physical model is an adaptation of Ref. [8], a 2-D model consisting of two

rows of N points (representing the dorsal and ventral sides of the worm). Each

point is acted on by the opposing forces of the elastic cuticle and pressure, as well

as muscle force and drag (often loosely referred to as friction or surface friction

[8]). We modify this model by introducing simpliﬁcations to reduce simulation

time, in part by allowing us to use a longer time step. Fig. 1A illustrates the

model’s structure. The worm is represented by a number rigid beams, connected

to each of the adjacent beams by four springs. Two horizontal (h) springs con-

nect points on the same side of adjacent beams and resist both elongation and

compression. Two diagonal (d) springs connect the dorsal side of the i

beam

to the ventral side of the i + 1

, and vice versa. These springs strongly resist

compression and have an eﬀect analogous to that of pressure, in that they help

to maintain reasonably constant area in each unit.

The model was implemented in C++, using a 4

order Runge-Kutta method

for numerical integration, with a time step of 0.1 ms.

Equations and parameters

The original model [8] required a time step of 0.001 ms with the same integration

method.

J.H. Boyle, J. Bryden, and N. Cohen

of the physical model are given in Appendix B. The steps taken to interface the

physical and neuronal models are described in Appendix C.

Results

Using our integrated model we ﬁrst simulated a single unit (the tail), and then

implemented two phase-lagged units (adding a body unit). In what follows, we

present these results, as compared to those of the neural model alone.

4.1

Single Oscillating Segment

The neural model alone produces robust oscillations in unit bending angle (θ

)

with a roughly square waveform, as shown in Fig. 2A. The model unit oscillates

at about 3.5 Hz, as compared to frequencies of about 0.5 Hz observed for C.

elegans forward locomotion on an agarose substrate. It has not been possible

to ﬁnd parameters within reasonable electrophysiological bounds for the neural

model that would slow the oscillations to the desired time scales [12].

Oscillations of the integrated neuro-mechanical model of a single unit are

shown in Fig. 2B. All but four parameters of the neuronal model remain un-

changed from Ref. [12]. However, parameters used for the actuation step caused

a slight asymmetry in the oscillations when integrated with a physical model,

and were therefore modiﬁed.

As can be seen from the traces in the ﬁgure, the frequency of oscillation

in the integrated model is about 0.5 Hz for typical agarose drag [8], and the

waveform has a smooth, almost sinusoidal shape. Faster (and slower) oscillations

are possible for lower (higher) values of drag. Fig. 2C shows a plot of oscillation

frequencies as a function of drag for the integrated model.

Fig. 2. Oscillations of, A, the original neural model [12] and, B, the integrated model

(with drag of 80

×10

−6

kg.s

−1

). Note the diﬀerent time scales. C: Oscillation frequency

as a function of drag. The zero frequency point indicates that the unit can no longer

oscillate.

An Integrated Neuro-mechanical Model of C. elegans Forward Locomotion

4.2

Two Phase-Lagged Segments

Parameters of the neural model are given in Table A-1 for the tail unit and in

Table A-2 for the body unit. Fig. 3 compares bending waveforms recorded from

a living worm (Fig. 3A), simulated by the neural model (Fig. 3B) and simulated

by the integrated model (Fig. 3C).

Fig. 3. Phase lagged oscillation of two units. A: Bending angles extracted from a

recording of a forward locomoting worm on an agarose substrate. The traces are of two

points along the worm (near the middle and

12

of a body length apart). B: Simulation

of two coupled units in the neural model. C: Simulation of the integrated model. Take

note of the faster oscillations in subplot B.

Discussion

C. elegans is amenable to manipulations at the genetic, molecular and neuronal

levels but with such rich behaviour being produced by a system with so few

components, it can often be diﬃcult to determine the pathways of cause and

eﬀect. Mathematical and simulation models of the locomotion therefore provide

an essential contribution to the understanding of C. elegans neurobiology and

motor control.

The inclusion of a realistic embodiment is particularly relevant to a model

of C. elegans locomotion. Sensory feedback is important to the locomotion of

all animals. However, in C. elegans, the postulated existence of stretch receptor

inputs along the body (unpublished communication, L. Eberly and R. Russel,

reported in [2]) would provide direct information about body posture to the

motor neurons themselves. Thus, the neural control is likely to be tightly coupled

to the shape the worm takes as it locomotes. Modelling the body physics is

therefore particularly important in this organism. Here we have presented the

ﬁrst steps in the implementation of such an integrated model, using biologically

plausible parameters for both the neural and mechanical components.

One interesting eﬀect is the smoothing of the waveform from a square-like

waveform in the isolated neural model to a nearly sinusoidal waveform in the

integrated model. The smoothing can be attributed to the body’s resistance to

bending (modelled as a set of springs), which increases with the bending angle.

J.H. Boyle, J. Bryden, and N. Cohen

By contrast, in the original neural model, the rate of bending depends only on

the neural output.

The work presented here would naturally lead to an integrated neuro-

mechanical model of locomotion for an entire worm. The next step toward this

goal, extending the neural circuit to the entire ventral cord (and the correspond-

ing motor system) is currently underway. The physical model introduces long

range interactions between units via the body and environment. In a real worm,

as in the physical model, for bending to occur at some point along the worm, lo-

cal muscles must contract. However, such contractions also apply physical forces

to adjacent units, and so on up and down the worm, giving rise to a signiﬁcant

persistence length. For this reason the extension of the neuro-mechanical model

from two to three (or more) units will not be automatic and will require param-

eter changes to model an operable balance between the eﬀects of the muscle and

body properties. In fact, the worm’s physical properties (and, in particular, the

existence of long range physical interactions along it) could set new constraints

on the neural model, or could even be exploited by the worm to achieve more

eﬀective locomotion. Either way, the physics of the worm’s locomotion is likely

to oﬀer important insights that could not be gleaned from a model of the isolated

neural subcircuit.

We have shown that a neural model developed with only the most rudimentary

physical framework can continue to function with a more realistic embodiment.

Indeed, both the waveform and frequency have been improved beyond what was

possible for the isolated neural model. We conclude that the body and environ-

ment are likely to be important components of the subsystem that generates

locomotion in the worm.

Acknowledgement

This work was funded by the EPSRC, grant EP/C011961. NC was funded by

the EPSRC, grant EP/C011953. Thanks to Stefano Berri for movies of worms

and behavioural data.

References

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2. White, J.G., Southgate, E., Thomson, J.N., Brenner, S.: The structure of the ner-

vous system of the nematode Caenorhabditis elegans. Philosophical Transactions

of the Royal Society of London, Series B 314, 1–340 (1986)

3. Ferr´

ee, T.C., Marcotte, B.A., Lockery, S.R.: Neural network models of chemotaxis

in the nematode Caenorhabditis elegans. Advances in Neural Information Process-

ing Systems 9, 55–61 (1997)

4. Ferr´

ee, T.C., Lockery, S.R.: Chemotaxis control by linear recurrent networks. Jour-

nal of Computational Neuroscience: Trends in Research, 373–377 (1998)

5. Wicks, S.R., Roehrig, C.J., Rankin, C.H.: A Dynamic Network Simulation of the

Nematode Tap Withdrawal Circuit: Predictions Concerning Synaptic Function Us-

ing Behavioral Criteria. Journal of Neuroscience 16, 4017–4031 (1996)

An Integrated Neuro-mechanical Model of C. elegans Forward Locomotion

6. Tsalik, E.L., Hobert, O.: Functional mapping of neurons that control locomotory

behavior in Caenorhabditis elegans. Journal of Neurobiology 56, 178–197 (2003)

7. Sakata, K., Shingai, R.: Neural network model to generate head swing in locomotion

of Caenorhabditis elegans. Network: Computation in Neural Systems 15, 199–216

(2004)

8. Niebur, E., Erd¨

os, P.: Theory of the locomotion of nematodes. Biophysical Jour-

nal 60, 1132–1146 (1991)

9. Niebur, E., Erd¨

os, P.: Theory of the Locomotion of Nematodes: Control of the

Somatic Motor Neurons by Interneurons. Mathematical Biosciences 118, 51–82

(1993)

10. Niebur, E., Erd¨

os, P.: Modeling Locomotion and Its Neural Control in Nematodes.

Comments on Theoretical Biology 3(2), 109–139 (1993)

11. Bryden, J.A., Cohen, N.: A simulation model of the locomotion controllers for

the nematode Caenorhabditis elegans. In: Schaal, S., Ijspeert, A.J., Billard, A.,

Vijayakumar, S., Hallam, J., Meyer, J.A. (eds.) Proceedings of the eighth interna-

tional conference on the simulation of adaptive behavior, pp. 183–192. MIT Press

/ Bradford Books (2004)

12. Bryden, J.A., Cohen, N.: Neural control of C. elegans forward locomotion: The role

of sensory feedback (Submitted 2007)

13. Bryden, J.A.: A simulation model of the locomotion controllers for the nematode

Caenorhabditis elegans. Master’s thesis, University of Leeds (2003)

14. Chen, B.L., Hall, D.H., Chklovskii, D.B.: Wiring optimization can relate neuronal

structure and function. Proceedings of the National Academy of Sciences USA 103,

4723–4728 (2006)

Appendix A: Neural Model

Neurons are assumed to have graded potentials [11,12,13]. In particular, motor

neurons (VB and DB) and are modelled by leaky integrators with a transmem-

brane potential V (t) following:

dV

dt

−G(V − E

rev

)

− I

shape

+ I

AVB

(A-1)

where C is the cell’s membrane capacitance; E

rev

is the cell’s eﬀective reversal

potential; and G is the total eﬀective membrane conductance. Sensory input

shape

j=1

− E

stretch

(θ

) is the stretch receptor input

from the shape of the body, where E

stretch

is the reversal potential of the ion

channels, θ

is the bending angle of unit j and σ

stretch

is a sigmoid response

function of the stretch receptors to the local bending. The stretch receptor ac-

tivation function is given by σ

stretch

(θ) = 1/ [1 + exp (

−(θ − θ

)/δθ)] where the

steepness parameter δθ and the threshold θ

are constants. The command input

current I

AVB

= G

AVB

− V ) models gap junctional coupling with AVB

(with coupling strength G

AVB

and denoting AVB voltage by V

AVB

). Note that

in the model, AVB is assumed to have a suﬃciently high capacitance, so that

the gap junctional currents have a negligible eﬀect on its membrane potential.

J.H. Boyle, J. Bryden, and N. Cohen

Segment bending in this model is given as a summation of an output function

from each of the two neurons:

dθ

dt

= σ

out

V B

(V )

− σ

out

(V ) ,

(A-2)

where σ

out

(V ) = ω

max

/[1 + exp (

−(V − V

0

)/δV )] with constants ω

max

, δV and

. Note that dorsal and ventral muscles contribute to bending in opposite di-

rections (with θ and -θ denoting ventral and dorsal bending, respectively).

Table A-1. Parameters for a self-oscillating tail unit (as in Ref. [12])

Parameter

Value

Parameter

Value

Parameter

Value

rev

−60mV

AVB

−30.7mV

5pF

19.07pS

17.58pS

AVB

35.37pS

AVB

13.78pS

stretch

98.55pS

stretch

67.55pS

stretch

60mV

0,VB

−18.68

0,DB

−19.46

δθ

0.1373

δθ

0.4186

max,VB

6987

/sec

max,DB

9951

/sec

0,VB

22.8mV

0,DB

25.0mV

δV

0.2888mV/sec

δV

0.0826mV/sec

Table A-2. Parameters for body units and tail-body interactions as in Ref. [12]. All

body-unit parameters that are not included here are the same as for the tail unit.

Parameter Value

Parameter

Value

Parameter

Value

26.09pS

25.76pS

stretch

16.77pS

stretch

18.24pS

stretch

60mV

0,VB

−19.14

0,DB

−13.26

δθ

1.589

/sec

δθ

1.413

/sec

Appendix B: Physical Model

The physical model consists of N rigid beams which form the boundaries between

the N

− 1 units. The i

beam can be described in one of two ways: either by

the (x, y) coordinates of the centre of mass (CoM

in Fig. 1) and angle φ

, or

by the (x, y) coordinates of its two end points (P

and P

in Fig. 1). Each

formulation has its own advantages and is used where appropriate.

B.1 Spring Forces

The rigid beams are connected to each of their neighbours by two horizontal (h)

springs and two diagonal (d) springs, directed along the vectors

k,i

= P

i+1

− P

for k = D, V

m,i

= P

i+1

− P

for k = D, V , l = V, D and m = 1, 2 ,

(B-1)

An Integrated Neuro-mechanical Model of C. elegans Forward Locomotion

for i = 1 : N

− 1, where P

= (x

, y

) are the coordinates of the ends of the i

beam. The spring forces F

(s)

depend on the length of these vectors, Δ

k,i

|Δ

k,i

and are collinear to them. The magnitude of the horizontal and diagonal spring

forces are piecewise linear functions

(s)

(Δ) =

⎧

⎪

⎨

⎪

⎩

(Δ

− L

) + κ

− L

)

Δ > L

(Δ

− L

)

> Δ > L

(Δ

− L

) + κ

− L

)

Δ < L

(Δ

− L

)

otherwise

(B-2)

(s)

(Δ) =

⎧

⎨

⎩

(Δ

− L

) + κ

− L

)

Δ < L

(Δ

− L

)

1

< Δ < L

0

0

otherwise

(B-3)

where spring (κ) and length (L) constants are given in Table B-1.

Table B-1. Parameters of the physical model. Note that values for θ

and θ

diﬀer

from Ref. [12] and Table A-1.

Parameter

Value

Parameter

Value

Parameter Value

80μm

50μm

0.5L

1.5L

+ D

0.95L

20μN.m

−1

10κ

0.5κ

10κ

50κ

10κ

muscle

0.005L

c = c

⊥

× 10

−6

kg.s

−1

0,VB

−29.68

0,DB

−8.46

0,VB

−22.14

0,DB

−10.26

B.2

Muscle Forces

Muscle forces F

(m)

are directed along the horizontal vectors Δ

k,i

with magnitude

(m)k,i

= f

muscle

k,i

for k = D, V and i = 1 : N

− 1 ,

(B-4)

where f

muscle

is a constant (see Table B-1) and A

k,i

are scalar activation functions

for the dorsal and ventral muscles, determined by

D,i

, A

V,i

) =

(θ

(t), 0)

(t)

≥ 0

(0,

−θ

(t))

(t) < 0 ,

(B-5)

where θ

(t) =

dθ

dt is the integral over the output of the neural model.

B.3

Total Point Force

With the exception of points on the outer beams, each point i is subject to

forces F

D,i

and F

V,i

, given by diﬀerences of the spring and muscle forces from

the corresponding units (i and i

− 1):

J.H. Boyle, J. Bryden, and N. Cohen

D,i

= (F

(s)D,i

− F

(s)D,i

−1

) + (F

(s)1,i

− F

(s)2,i

−1

) + (F

(m)D,i

− F

(m)D,i

−1

)

V,i

= (F

(s)V,i

− F

(s)V,i

−1

) + (F

(s)2,i

− F

(s)1,i

−1

) + (F

(m)V,i

− F

(m)V,i

−1

) .

(B-6)

Since the ﬁrst beam has no anterior body parts, and the last beam has no

posterior body parts, all terms with i = 0 or i = N are taken as zero.

B.4

Equations of Motion

Motion of the beams is calculated from the total force acting on each of the

2N points. Since the points P

and P

are connected by a rigid beam, it is

convenient to convert F

(t)k,i

to a force and a torque acting on the beam’s centre

of mass.

Rotation by φ

converts the coordinate system of F

(t)k,i

= (F

(t)k,i

, F

(t)k,i

)

to a new system F

(t)k,i

= (F

⊥

(t)k,i

, F

(t)k,i

) with axes perpendicular to (

⊥) and

parallel with ( ) the beam:

⊥

(t)k,i

= F

(t)k,i

cos(φ

) + F

(t)k,i

sin(φ

)

(t)k,i

= F

(t)k,i

cos(φ

)

− F

(t)k,i

sin(φ

) .

(B-7)

The parallel components are summed and applied to CoM

, resulting in pure

translation. The perpendicular components are separated into odd and even

parts (giving rise to a torque and force respectively) by

⊥,even

⊥

(t)D,i

+ F

⊥

(t)V,i

)

⊥,odd

⊥

(t)D,i

− F

⊥

(t)V,i

)

(B-8)

As in Ref. [8] we disregard inertia, but include Stokes’ drag. Also following Ref.

[8], we allow for diﬀerent constants for drag in the parallel and perpendicular

directions, given by c and c

⊥

respectively. The motion of CoM

is therefore

(CoM),i

(t)D,i

+ F

(t)V,i

)

⊥

(CoM),i

⊥

(2F

⊥,even

)

(CoM),i

⊥

(2F

⊥,odd

) ,

(B-9)

where r = 0.5D is the radius of the worm. Finally we convert V

(CoM),i

and

⊥

(CoM),i

back to (x, y) coordinates with

(CoM),i

= V

(CoM),i

cos(φ

)

− V

⊥

(CoM),i

sin(φ

)

(CoM),i

= V

⊥

(CoM),i

cos(φ

) + V

(CoM),i

sin(φ

) .

(B-10)

An Integrated Neuro-mechanical Model of C. elegans Forward Locomotion

Appendix C: Integrating the Neural and Physical Model

In the neural model, the output dθ

(t)/dt speciﬁes the bending angles θ

(t) for

each unit. In the integrated model, θ

(t) are taken as the input to the muscles.

Muscle ouputs (or contraction) are given by unit lengths. The bending angle α

is then estimated from the dorsal and ventral unit lengths by

= 36.2

|Δ

D,i

| − |Δ

V,i

(C-1)

where L

is the resting unit length. (For simplicity, we have denoted the bending

angles of both the neural and integrated models by θ in the Figures).

M. Ishikawa et al. (Eds.): ICONIP 2007, Part I, LNCS 4984, pp. 48–53, 2008.

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