Nonlinear Model for Complex Neurons in Biological Visual Visions

Sasan Mahmoodi

and Nasim Saba

Electronic and Computer Science, University of Southampton, Building 1, University Road, SO17 1BJ,

Southampton, Hampshire, U.K.

Keywords: Complex Neurons, Biological Visual Systems, Nonlinear Systems, Nonlinear Cells.

Abstract: Complex cells in biological visual vision are well known to be nonlinear. In this paper, it is demonstrated that

these nonlinear complex cells can be modelled under some certain conditions by a biologically inspired model

which is nonlinear in nature. Our model consists of cascaded neural layers accounting for anatomical evidence

in biological early visual visions. In the model proposed in this paper, the axons associated with the complex

cells are considered to operate nonlinearly. We also consider the second order interaction receptive maps as

directional derivatives of the complex cell's kernel along the direction of orientation tuning. Our numerical

results are similar to the biologically recorded data reported in the literature.

1 INTRODUCTION

The concept of visual receptive fields is introduced in

(Hartline 1938) as a region in visual field in which if

visual stimuli are presented, the corresponding cell

responds. The sub-regions associated with ON and

OFF responses are then discovered in (Kuffler, 1953).

Hubel and Wiesel introduce the orientation tuning of

neurons in the primary visual cortex (Hubel and

Wiesel, 2005). The receptive mapping techniques

based on white noise stimuli are then exploited in

(DeAngelis et al., 1995; DeAngelis and Anzai, 2004).

Motion perception based on energy models is also

investigated in (Adelson and Bergen, 1985) by using

oriented filters in the space-time domain. In fact,

biological experiments quantitatively indicate that the

linear visual receptive fields are Gaussian-related

kernels. In a mathematical setting, scale-space theory

presents a general framework for early visual systems

by postulating a set of axioms which an early visual

system is expected to possess. Such a framework then

leads to Gaussian-related kernels characterizing any

linear visual system including early biological visual

systems when they behave linearly (see e.g. Weickert

et al., 1999; Lindeberg, 2011; Lindeberg, 2013; ter

Haar Romeny et al., 2001; ter Haar Romeny, 2003;

Koenderink, 1988; Florack, 1997). On the other hand,

a model based on the anatomical and physiological

properties of biological visual systems is proposed in

(Mahmoodi, 2015) to derive Gaussian-related kernels

in spatial as well as spatio-temporal domains. The

model presented in (Mahmoodi, 2015) is not linear in

nature. Therefore the conditions under which this

system become linear is discussed in (Mahmoodi,

2015). Under such conditions, linear Gaussian related

filters are derived (Mahmoodi, 2015). In such a

model, the functionalities of Lateral Geniculate

Nucleus (LGN) cells and simple cells such as linear

isotropic separable, non-isotropic separable and non-

separable (velocity-adapted) cells, with Gaussian

related receptive fields can be explained (Mahmoodi,

2015). In this paper, the nonlinearity of this model is

also considered and it is demonstrated that under

certain conditions, the behaviour of nonlinear

complex cells may be attributed to this non linearity

of the model. Here our contribution is to explain the

nonlinear nature of complex cells by using the

nonlinear model of early visual system proposed in

(Mahmoodi, 2015). We also demonstrate that the

second order interactions of receptive maps for

complex cells may be explained as directional

derivatives of the neuron's kernel along the direction

of orientation tuning. The structure of the rest of the

paper is as follows. In section 2, our nonlinear model

is explained. Section 3 presents the numerical

analyses and results and finally conclusions are drawn

in section 4.

2 MODEL

The nonlinear model proposed in (Mahmoodi, 2015)

162

Mahmoodi, S. and Saba, N.

Nonlinear Model for Complex Neurons in Biological Visual Visions.

DOI: 10.5220/0005692601620167

In Proceedings of the 9th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2016) - Volume 4: BIOSIGNALS, pages 162-167

ISBN: 978-989-758-170-0

Copyright

c

2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

behaves linearly under some certain conditions. The

linear mode of this nonlinear model is therefore fully

investigated in (Mahmoodi, 2015). Linear Gaussian-

related kernels associated with linear Lateral

Geniculate Nucleus (LGN) cells and simple cells in

strait cortex are derived for when the model behaves

linearly. Complex cells on the other hand are

nonlinear. Here we exploit the nonlinearity of the

model presented in (Mahmoodi, 2015) to explain the

nonlinear behaviour of the complex cells. According

to the model presented in (Mahmoodi, 2015), neurons

in early biological visual system are connected in

layers which are cascaded from retina to striate

cortex.

A layer of neurons is shown in Figure (1-top).

These neurons are cascaded in a way that the axons

of the neurons in the previous layer are connected to

the dendrites of the neurons in the current layer. A 2D

illustration of these cascaded layers are depicted in

figure (1-bottom).

Each horizontal line in this figure, represents a

neural layer and vertical layers represent axons to

connect a layer (a horizontal line) to the next one

(another horizontal line above the previous line). It

is believed that most complex cells have a linear-

nonlinear (L-N) structure. The linear part of these

cells is simply a linear Gaussian-related kernel

(DeAngelis and Anzai, 2004). The first step is

therefore to find a formula to explain the input-output

relationship of the nonlinear part of these complex

cells. Axons transmitting neural spikes from a neuron

to another one are modelled as transmission lines. If

neuron A sends n spikes through its axon to neuron

B. The potential received in the dendrites of neuron B

is calculated as (see Mahmoodi, 2015 for more

details):

N

n

N

n

nn

h

nho

aTT

t

tzv

aTtzvtzw

11

)exp(

2

),(

)exp(),(),(

(1)

where

n

T

= the time the nth spike is released from

neuron A

t = the time that a spike reaches to neuron B from

neuron A,

N = the total numbers of spikes,

z = the length of axon,

t

zCR

C

tG

CRt

tzv

zz

z

z

zz

h

4

expexp

1

2

1

),(

2

z

G

,

z

C

, and

z

R

= conductance, capacitance, and

resistance of the axon (transmission line) per unit

length,

and finally

2

2

4t

zCR

C

G

a

zz

z

z

.

Figure 1: (top) the configuration of neurons in a layer with

respect to spatial coordinates x and y (bottom) the 2D

representation of the configuration of neurons in the

cascaded layers of neurons.

n

T

can be written as the summation of all time

intervals between consecutive spikes, i.e.:

n

m

mn

TnTT

1

(2)

where

T

is the average time interval between

consecutive spikes. Let us consider the case where a

series of spikes are transmitted through axon from

neuron A to neuron B in a time period much less than

the time required for spikes to travel from neuron A

to neuron B, i.e.

tT

n

)( max

n

. By replacing (2) in

(1) and assuming that

tT

n

)( max

n

and therefore

ignoring the second term in (1), one can write:

x-axis

x

-axis

Flow

of

Visual

Signal

y axis

Nonlinear Model for Complex Neurons in Biological Visual Visions

163

N

n

ho

Tantzvtzw

1

)exp(),(),(

(3)

Equation (3) can be rewritten as:

1

1

)exp(),(),(

e

e

Tatzvtzw

N

ho

(4)

It is also reasonable to assume the average time

interval between consecutive spikes is too small, i.e.

1Ta

. Equation (4) therefore is approximated as:

1

1

),(),(

e

e

tzvtzw

N

ho

(5)

According to classical rectification model for neural

firing rate (Carandini and Fester, 2000), the input

potential of neuron A is proportional to firing rate and

therefore N. Equation (5) determines the input-output

relationship of a cell behaving nonlinearly. For small

values of potentials, N is very small and therefore

0),( tzw

o

According to classical rectification model

(Carandini and Fester, 2000), N increases linearly

with respect to input potential

in

V

, i.e.

in

kVN

Therefore for large positive potentials, the input-

output relationship of such a nonlinear cell will look

like a half rectified power function as reported in

(DeAngelis and Anzai, 2004) and shown in figure (2-

bottom).

3 NUMERICAL RESULTS

Figure (2-top) shows the input-output relationship for

a nonlinear cell according to equation (5) and as an

example for k=5, i.e. for

in

VN 5

. The similarity

between figure (2-top) and figure (2-bottom) as

reported in (DeAngelis and Anzai, 2004) is

interesting and important.

The linear part of a complex cell is simply the sum

of three cells with linear isotopic kernels such as the

one shown in figure (1-bottom).The outputs of these

three cells are summed by another cell whose axon

behave nonlinearly governed by equation (5), i.e.:

),()

2

)(2

(),(

2

2

),(

2

)(2

),(),,(

1

1

tzv

t

yxh

t

yxh

t

yxhtyx

nn

n

(6)

where

),( yxh

n

is the isotropic kernel for layer n and

)(t

is the impulse response of the cells' axons

behaving linearly (Mahmoodi, 2015). The outputs of

these three cells are summed by another cell

according to equation (6).

Figure 2: (top) The input-output relationship calculated

based on equation (5) and for k=5, (bottom) Biologically

recorded input-output relationship according to (DeAngelis

and Anzai 2004).

The impulse response of the axon behaving

nonlinearly for this cell is represented by

),( tzv

. The

spatio-temporal profile of a complex cell consisting

of three simple cells with linear isotropic kernels for

a single stimuli (nonlinear case) is shown in figure (3-

top). This is similar to the biological recorded data

shown in figure (3-bottom) in (DeAngelis and Anzai,

2004).

The response of this nonlinear complex cell for

the case where two simultaneous and spatially

separated stimuli (second order interactions) are

presented in the visual field, is shown in figure (4-

top).On the other hand, there are some other complex

cells described by equation (7):

),(),(),(),(),,( tzvyxhyxhyxhtyx

x

n

x

n

x

n

(7)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

50

100

150

200

250

Input Potential

Output Potential

BIOSIGNALS 2016 - 9th International Conference on Bio-inspired Systems and Signal Processing

164

Figure 3: (top) The spatio-temproal profile of a nonlinear

complex cell for a single stimulus. (bottom) The

biologically recorded spatio-temporal profile according to

(DeAngelis and Anzai, 2004).

The neural configuration of equation (7) is

depicted in figure (1-top). In this configuration, the

nonlinear-behaving axon of the cell summing the

outputs of the three cells is represented by

),( tzv

in

equation (7). In this paper, we hypothesize that the

receptive map in the case of the second order

interactions is equivalent to directional derivatives

along the direction of the orientation tuning of the

kernel

),,( tyx

in equation (6) or (7). For a complex

cell whose kernel is represented by equation (7), this

second order interactions is described by:

)(sin

)2sin()(cos

2

2

2

2

2

2

2

2

2

y

yx

xn

(8)

where

)sin()cos(

jin

is along the

direction of orientation tuning of the cell. The

directional derivative of equation (8) for

o

45

is

calculated in figure (4-top).

Figure 4: (top) A nonlinear receptive field map calculated

by using equations (7) and (8) (bottom) the biologically

recorded second order interactions of a receptive map for a

complex cell as reported in (DeAngelis and Anzai, 2004).

The similarity between the map calculated in

figure (4) and the biologically recorded data in figure

(4-bottom) as reported in (DeAngelis and Anzai,

2004) is interesting.

A fourth order directional derivative in space x

and time t for a complex cell represented by equation

(6) is calculated in figure (5-top). The biologically

recorded result shown as the second order interaction

of a spatio-temporal profile of a complex cell shown

in figure (5-bottom) as reported in (DeAngelis and

Anzai, 2004) is similar to our result depicted in figure

(5-top).

Nonlinear Model for Complex Neurons in Biological Visual Visions

165

Figure 5: (top) Fourth order directional derivative of a

spatio-temporal (x-t) receptive map, (bottom) second order

interactions for x-t receptive map of a complex cell as

reported in (DeAngelis and Anzai, 2004).

As can be seen from figure (5-top), there are five

lobes (three positive and two negative lobes) among

which the middle lobe is the strongest. This is similar

to the biologically recorded data shown in figure (5-

bottom) as reported in (DeAngelis and Anzai, 2004).

This second order interaction map is reminiscent of

the space-time inseparable linear maps predicting the

direction selectivity of complex cells. It is noted that

the fourth order directional derivatives should be

calculated along the tilt of space-time response

pattern.

4 CONCLUSIONS

A nonlinear model based on the model presented in

(Mahmoodi, 2015) to explain the nonlinear behaviour

of complex cells is proposed here. According to our

model, the nonlinearity of the complex cells may be

routed from the fact that the axons of neurons behave

nonlinearly under certain conditions. These

conditions are explained here. In this paper, some

approximations to this nonlinear model of neurons are

made to demonstrate that the complex cells behave

like half rectified power functions corresponding to

biologically recorded data. It is then shown here that

the space-time calculated receptive map in our model

for the complex cells when a single stimulus is

presented to the neuron, is similar to the biological

receptive field of the complex cells. We then

hypothesize that the second order interactions of

complex cells recorded in biology may be equivalent

to the directional derivatives of the visual receptive

map of the complex cells. Our results demonstrate

that the directional derivatives of the space or space-

time visual receptive maps of complex cells show

similar response patterns to the biologically recorded

second order interactions confirming our hypothesis.

REFERENCES

Adelson E, Bergen J 1985. Spaio-temporal Energy Models

for the Perception of Motion. Journal of Optical Society

of America A, Vol. 2, pp. 284-299.

Carandini M, Ferster D 2000. Membrane Potential and

Firing Rate in Cat Primary Visual Cortex. Journal of

Neuroscience, Vol. 20, No. 1, pp. 470-484.

DeAngelis GC, Anzai A 2004. A Modern View of the

Classical Receptive Field: Linear and non-Linear

Spatio-temporal Processing by V1 Neurons. In L.M.

Chalupa, and J.S. Werner (eds.), The Visual

Neurosciences, Vol 1, MIT Press, Cambridge, pp.

704-719.

DeAngelis DC, Ohzawa I, Freeman RD 1995. Receptive-

field Dynamics in the Central Visual Pathways. Trends

in Neurosciences, Vol. 18, pp:451-457.

Florack, L 1997. Image Structure. Series in Mathematical

Imaging and Vision, Springer, Dordrecht.

Koenderink JJ 1988. Scale-Time, Biological Cybernetic,

Vol. 58, pp. 159-162.

Hartline HK 1938. The Response of Single Optic Nerve

Fibres of the Vertebrate Eye to Illumination of the

Retina. American Journal of Physiology, Vol. 121, pp.

400-415.

Hubel DH, Wiesel TN 2005. Brain and Visual Perception:

the Story of a 25-year collaboration. Oxford University

Press, Oxford.

Kuffler SW 1953. Discharge Patterns and Functional

Organization of Mammalian Retina. Journal of

Neurophysiology, Vol. 16, No. 1, pp. 37-68.

Lindeberg T 2011. Generalized Gaussian Scale-Space

Axiomatic Comprising Linear Scale-Space, Affine

Scale-Space and Spatio-Temporal Scale-Space.

Journal of Mathematical Imaging and Vision, Vol. 40,

pp: 36-81.

Lindeberg T 2013. A Computational Theory of Visual

Receptive Fields. Biological Cybernetics, Vol. 107, pp:

BIOSIGNALS 2016 - 9th International Conference on Bio-inspired Systems and Signal Processing

166

589-635.

Mahmoodi, S., 2015. Linear Neural Circuitry Model for

Visual Receptive Fields. Journal of Mathematical

Imaging and Vision.

ter Haar Romeny B 2003. Front-End Vision and Multi-

Scale Image Analysis. Springer Dordrecht.

ter Haar Romeny B, Florack L, Nielsen M 2001. Scale-time

Kernels and Models, In: Scale-Space and Morphology,

Proceedings of Scale-Space’01, Vancouver, Canada,

LNCS, Springer, Berlin.

Weickert J, Ishikawa S, Imiya A (1999) Linear Scale Space

has First been proposed in Japan, Journal of

Mathematical Imaging and Vision, Vol. 10, pp. 237-

252.

Nonlinear Model for Complex Neurons in Biological Visual Visions

167