Design of Neuromorphic Hardwareskohno/doc/ohpe050825.pdfProperties of biological neurons (4/7) What...

Design of NeuromorphicHardwares

Takashi Kohno†

[email protected]

† Aihara Complexity Modelling Project, ERATO, JST,,Japan.

Introduction (0)Aihara Complexity Modelling Project

The first group formulates and analyzesmathematical models on biological systems.

e.g. neural and genetic networks

The second group analyzes epidemics of emerginginfectious diseases.

e.g. SARS and influenza

The third group explores in new kind of computationby complex systems.

e.g. neuromorphic hardwares and chaotic random numbergenerators

CCA2005 Satellite Seminar 2005.08 – p.1/69

Introduction (1/4)Contents

IntroductionWhat does “neuromorphic hardware” denote?A classical example of neuromorphic hardware.

Silicon neuronWhy silicon neuron is studied?Properties of biological neuronsModels for biological neuronsProperties of MOSFETConventional design principles for silicon neuronMathematical-model-based design

SummaryCCA2005 Satellite Seminar 2005.08 – p.2/69

Introduction (2/4)What does “neuromorphic hardware” denote?

Definition :Device designed to reproduce some

properties in neural systems.

Applicable studiesSilicon retina, Electronic cochlea, Selective attention

system(SAS), ...

Silicon neuronsI&F neuron, Small neural network, Neural prosthesis, ...


Introduction (3/4) - Silicon retina -

Example : Silicon retinaEmulated structures andfunctions in biological retina:

Photoreceptor:converts illuminational input toelectric signal (logarithmic).

Horizontal cell:generates spacial and temporalaverage of photoreceptor outputs.

Bipolar cell:detects the difference between thephotoreceptor and the horizontal cellsignals.

OCCA2005 Satellite Seminar 2005.08 – p.4/69


Example : Silicon retinaEmulated structures andfunctions in biological retina:

Photoreceptor:converts illuminational input toelectric signal (logarithmic).

Horizontal cell:generates spacial and temporalaverage of photoreceptor outputs.

Bipolar cell:detects the difference between thephotoreceptor and the horizontal cellsignals.



Structure of silicon retina

Reproduced properties


Silicon neuronWhy silicon neuron is studied?

Properties of biological neurons

Models for biological neurons

Properties of MOSFET

Conventional design principles for silicon neuron

Mathematical-model-based design


Why silicon neuron is studied? (1/2)Nerve system

highly parallelized structure

flexible in many aspects

robust and complex process-ing realized by unreliable andsimple devices

NeuronNeuron is the universalconstituent element in nervesystem.

Operation principle of nervesystem is not elucidatedcompletely.


Why silicon neuron is studied? (2/2)Significance of studies on silicon neuron

Validation of theoretical studiesimplementability, effect of real-world noise, ...

Simulationacceleration for large scale network of complex models

Applicationneural prosthesis, system control, pattern recognition, ...

Start point forconstructing genuinely neuromorphic systems


Properties of biological neurons (1/7)Simplified illustration of neuronal cell

Dendrites“input terminals”

Axon hillockcenter of informationprocessing

Axon“output terminals”


Properties of biological neurons (2/7)Cell with excitability

fluidconsists of ions, ATPs,hormones, ...

cell membranecomposed of lipids andrepels ions

ionic channelspassive pathways ofspecific ions

ATPasesactive pathways of ions


Properties of biological neurons (3/7)Ionic concentrations in intracellular and extracellular fluids

(mM/kgH2O)

squid axon mammalian muscular cell

ion intracellular extracellular intracellular extracellular

K+ 400 20 155 4Na+ 50 440 12 145Cl− 40 ∼ 150 560 4 120A− 385 155

The ionic concentrations are different betweenintracellular and extracellular fluids.Na+, Cl− : dense in extracellular fluidK+ : dense in intracellular fluid


Properties of biological neurons (4/7)What keeps the ionic concentration difference against

leakage along the ionic channels?

Constant field assumption:Difference in electrical potentials cancels one in

concentration potentials.

Goldman-Hodgkin-Katz equation:mathematical model for membrane potential under existence ofplural ions






Membrane potential:voltage potential of intracellular fluid against extracellular one



Properties of biological neurons (5/7)Goldman-Hodgkin-Katz equation:

Em = RTF

lnP

ip+

i a+(o)i +

P

jp−

j a−(i)j

P

ip+

i a+(i)i +

P

jp−

j a−(o)j

Em : membrane potential (potential of intracellular fluid against extracellular one),

p±i : ionic permeability through the cell membrane for ion i,

F : Faraday constant, T : absolute temperature, R : gas constant,

a±(o)i : effective concentration of positive/negative ion i in the extracellular fluid,

a±(i)i : effective concentration of positive/negative ion i in the intracellular fluid

Equilibrium potential for ion i :

Em value in the case the fluids contain a single variety of ions i.

If ionic permeability for a specific ion i is absolutely higher thanthe others, Em approaches to it.


Properties of biological neurons (6/7)Membrane potential

Resting membrane potential:The membrane potential is constant without any stimuli in

many neurons. -65mV in squid axon

Action potential:The membrane potential overshoots drastically when

some stimuli are given.

Zeeman’s characterization (1971)

[overshoot] absolutely larger than stimulus

[threshold] overshoot emerges only when

stimulus is strong enough

[refractoriness] threshold increases after

overshoot






some stimuli are given.Zeeman’s characterization (1971)





overshoot


Properties of biological neurons (7/7)Action potential is

believed to be the media for neurons’ informationprocessing.

generated by the dynamical fluctuation of ionicpermeabilities of cell membrane.

What kind of laws determine the behaviorof ionic permeabilities?


Models for biological neurons (1/16)The most classical and basic model

Hodgkin-Huxley equations (1952):

CdEm

dt= gNam3h(ENa − Em)

+ gKn4(EK − Em) + gL(EL − Em)

dm

dt= αm − (αm + βm)m

dh

dt= αh − (αh + βh)h

dn

dt= αn − (αn + βn)n

ENa, EK , EL : equilibrium potentials for sodium, potassium,

and remaining ions respectively, gNa ≈ 120(mS/cm2),

gK ≈ 36(mS/cm2), gL ≈ 0.3(mS/cm2), C ≈ 1(µF/cm2)


Models for biological neurons (2/16)sodium channel parameters potassium channel parameter

dm

dt= αm − (αm + βm)m

αm =0.1(y + 40)

1 − exp(−(y + 40)/10)

(αm = 1 when y = −40)

βm = 4exp(−y + 65

18)

dh

dt= αh − (αh + βh)h

αh = 0.07exp(−y + 65

20)

βh =1

1 + exp(−(y + 35)/10)

dn

dt= αn − (αn + βn)n

αn =0.01(y + 55)

1 − exp(−(y + 55)/10)

(αn = 1 when y = −55)

βn = 0.125exp(−y + 65

80)


Models for biological neurons (3/16)

nullclines time constants

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-100 -80 -60 -40 -20 0 20 40 60 80 100

m, h

, n

Em [mV]

dm/dt=0dh/dt=0dn/dt=0

0

1

2

3

4

5

6

7

8

9

10

-100 -80 -60 -40 -20 0 20 40 60 80 100Tm

, Th,

Tn

Em [mV]

TmThTn

sigmoidal shape

range from 0 to 1

Tm is always more than an or-der of magnitude smaller thanTh and Tn.


Models for biological neurons (4/16)Generalized Hodgkin-Huxley formalism

Cydy

dt= − Iion(y, w1, · · · , wn) + Istim

dwi

dt=

wi,∞(y) − wi

Ti(y)

y : membrane potential, Cy : membran capacitance,Istim : stimulus input,wi : ionic channel activity parameters, wi,∞ : nullcline for wi,Ti : time constant for wi,

Most of the successive excitable membrane modelsconform to this formalism.


Models for biological neurons (5/16)The more tractable model

Morris-Lecar model (1981):

Cdy

dt= −gCam(y − ECa)

− gKw(y − EK) − gL(y − EL) + I

dm

dt= φ

(m∞(y) − m)

τm(y)

dw

dt= φ

(w∞(y) − w)

τw(y)

y : membrane potential, φ : temperature factor,

ECa, EK , EL : equilibrium potentials for calcium, potassium,

and remaining ions respectively,

gK ≈ 8(mS/cm2), gL ≈ 2(mS/cm2), C ≈ 20(µF/cm2)


Models for biological neurons (6/16)Reduced Morris-Lecar equations:

τm(y) << τw(y) ⇒ m → m∞(y)

Cdy

dt= −gCam∞(y)(y − ECa)

− gKw(y − EK) − gL(y − EL) + I

dw

dt= φ

(w∞(y) − w)

τw(y)

Two-variable model can be treated in phase plane.


Models for biological neurons (7/16)

calcium channel parameter potassium channel parameter

m = m∞(y)

m∞(y) =1

2(1 + tanh (

y − V1

V2))

dw

dt= φ

(w∞(y) − w)

τw(y)

w∞(y) =1

2(1 + tanh (

y − V3

V4))

τw(y) = 1/ cosh (y − V3

2V4)

V1 = −1.2(mV ), V2 = 18(mV ) Two typical parameter sets:

type 1 V3 = 12(mV ), V4 = 17.4(mV ),

gCa = 4.0(mS/cm2), φ = 1/15

type 2 V3 = 2(mV ), V4 = 30(mV ),

gCa = 4.4(mS/cm2), φ = 0.04


Models for biological neurons (8/16)Nullclines of Morris-Lecar equations:

I = gCam∞(y)(y − ECa) + gKw(y − EK) + gL(y − EL) · · · y-nullcline

w = w∞(y) · · ·w-nullcline

A phase plane of the type 1 setting

(S), (T), (U) are stable,saddle, and unstableequilibrium points,respectively.

The stable manifolds of(T) act as separatrices,and give the threshold.


Models for biological neurons (9/16)A phase plane of the type 2 setting

(S) is the unique stableequilibrium point.

The ascending limb ofthe y-nullcline isunstable, and givesthreshold.

The threshold phe-nomenon is less steepthan the type 1 setting.


Models for biological neurons (10/16)Neural excitability

Being stimulated with a sustainedstimulus Ia :

Class 1 neurons begin to firerepetitively with arbitrarily lowfrequency.

Class 2 neurons begin to firerepetitively with nonzerofrequency.

Functions in neural network:

Class 1 neurons may act as leaky integrators.

Class 2 neurons may act as resonators.

N. Masuda and K. Aihara, Proc. SBRN 2004, E. M. Izhikevich, International J. Bif. Chaos, 2000


Models for biological neurons (11/16)Class 2 neural excitability

Subcritical Hopf bifurcation on the rest state pointcan produce Class 2 neural excitability.

e.g. The M-L eqs. in the type 2 parameterset, the H-H eqs., ...

An equilibrium loses(gains) stability withnonzero imaginaryparts of the eigenval-ues.

The Hopf bifurcation theory guarantees that an unstablelimit cycle exists around the stable equilibrium.



Subcritical Hopf bifurcation on the rest state pointcan produce Class 2 neural excitability.

e.g. The M-L eqs. in the type 2 parameterset, the H-H eqs., ...

An equilibrium loses(gains) stability withnonzero imaginaryparts of the eigenval-ues.

The Hopf bifurcation theory guarantees that an unstablelimit cycle exists around the stable equilibrium.


Models for biological neurons (12/16)Class 2 excitability - phse plane example -

Stimulus current I = 90 (µA/cm2) is given to the type 2 setting:

The y-nullcline shifts up.

The unique equilibriumloses stability via a Hopfbifurcation.

An unstable limit cycleexists around the equilib-rium.


Models for biological neurons (13/16)Class 2 excitability - bifurcation diagram -

A subcritical Hopfbifurcation generates anunstable limit cycle.

The system state jumpsto the stable limit cyclewhen I exceeds theHopf bifurcation point.

⇓

The system begins tooscillate in nonzerofrequency.



Saddle-node on invariant circle bifurcation canproduce Class 1 neural excitability.

a) (S): a stable point,(T): a saddle point.

(S) and (T) approach each other

b) (S) merge with (T).An unstable manifold of (T) van-

ishes. The remaining one be-

comes a homoclinic manifold.

c) The homoclinic manifold becomes a stable limit cycle.The nearer to b) the system is, the longer the period of the limit cycle is.



Saddle-node on invariant circle bifurcation canproduce Class 1 neural excitability.

a) (S): a stable point,(T): a saddle point.

(S) and (T) approach each other

b) (S) merge with (T).An unstable manifold of (T) van-

ishes. The remaining one be-

comes a homoclinic manifold.




Stimulus current I is increased from 30 to 50(µA/cm2)in the type 1 setting:


(S) and (T) merge eachother and vanish.(Saddle-node bifurcation)

An unstable manifold of(T) turns to a homoclinicmanifold and then to a sta-ble limit cycle.


Models for biological neurons (16/16)Class 1 excitability - bifurcation diagram -

A saddle-node bifurcationgenerates an stable limitcycle. (when I = 40 (µA/cm2))

The system state transitto a stable limit cycle viathe homoclinic orbit atthe bifurcation point.

⇓

The system begins tooscillate in arbitrarilylow frequency.


Properties of MOSFET (1/3)MOSFET (enhancement)

A widely used electrical device.⇒ well developed design technics

Voltage-driven and current-driving.⇒ facility to design, low power consumption

Fabrication technologies are matured.⇒ high reliability

Small in size.⇒ high density integration in VLSIs


Properties of MOSFET (2/3)Characteristics of MOSFETs [Quadratic curve]

n channel MOSFET

Id = β2(Vi − θ)2

β : transconductance coefficient

θ : threshold voltage

p channel MOSFET

Id = −β2(Vi − θ)2




Properties of MOSFET (3/3)A simple MOSFET circuitry [Sigmoidal curve]

Differential pair circuitry with n channel MOSFETs

Id1 = Icmn

2+ β

4Vi

√

4Icmn

β− V 2

i

Id2 = Icmn

2− β

4Vi

√

4Icmn

β− V 2

i

when V 2i ≤ 2Icmn

β

β : transconductance coefficient, Icmn : constant bias current

Output currents are dependent on thevoltage difference between the two inputterminals.

Sigmoidal characteristics curves.



Properties and models of biological neurons

Properties of MOSFETs

Conventional design principles for silicon neuronIntroductionPhenomenological designsConductance-based designs

Mathematical-model-based design


Conventional design principlesTwo conventional and one new design principles for

silicon neurons

Phenomenological designsreproduce some phenomena in biological neurons.

e.g. Integrate-and-Fire neurons .

Conductance-based designsreproduce dynamics of the ion channels on the nervemembranes.

e.g. Silicon implementations of H-H eqs., M-L eqs., and so on.

Mathematical-model-based designsreproduce phase portrait structures of biologicalneuron models.


Phenomenological designs (1/4)Concept

Simple implementation.

The properties not regarded may not be inherited.⇒ Suitable only for the supposed applications.

High design efforts are needed for additionalproperties.⇒ Complete renewal of the circuit design may be required,

especially for optimized circuits.


Phenomenological designs (1/4)Concept

Simple implementation.

The properties not regarded may not be inherited.⇒ Suitable only for the supposed applications.

High design efforts are needed for additionalproperties.⇒ Complete renewal of the circuit design may be required,

especially for optimized circuits.CCA2005 Satellite Seminar 2005.08 – p.37/69

Phenomenological designs (2/4)An implementation of Leaky-Integrate and Fire (I&F) neuron

Bases on C. Mead, 1989G. Indiveri, Proc. IEEE Int. Symp. Circuits

and Systems-IV, 2003

Compact circuitry.(20 MOSFETs)

Low power consumption byoperation in subthreshold re-gion of MOSFETs. (≈ 1µW)

Reproduces:Integration property of spatiotemporal inputs,Threshold property of generating action potentials,Refractoriness after action potential generations, andSpike-frequency adaptation.


Phenomenological designs (3/4)An implementation of Leaky-I&F neuron

A waveform example Freq.-Stimulus curve.

Vmem increases linearly until overshoot and reset.

Class 1 excitability: Vrfr ↑ ⇒ refractoriness↓ ⇒ spike frequency↑


Phenomenological designs (4/4)An implementation of Leaky-I&F neuron

Response to sustained current stimulus.

Firing frequency decreasesas the spiking proceeds.


Conductance-based designs (1/5)Concept

Circuits that have the same dynamics as the ionicconductances are pursued.

⇒ Additional properties can be built in with theadditional membrane conductances.

⇒The circuitries tend to be complex, especiallyif the models are not suitable for FET imple-mentations.


Conductance-based designs (2/5)Example : Silicon neuron by Simoni et al.

Their aim is to construct hybrid systems.

Electrical devices connected to living neurons.

Implantable biomedical devices.

Composed of small silicon neuronal networks.e.g. Central pattern generator (CPG)

Their requirements are:

faithful reproduction of the ionic dynamics,

real-time operation,

appropriate signal level, and

compact size, low-power consumption.


Conductance-based designs (3/5)Silicon neuron by Simoni et al.

The ionic dynamics in the leech heart interneuron isreproduced.

consists of 7 ionic currents

C dVdt

= INa + IP + IK1 + IK2 + ICa + Ih + Ileak + Iinj

Ileak = gleak(Eleak − V ) Ij = gjmκj hj(Ej − V )

dmj

dt=

1

τmj(V )

(m∞j(V ) − mj)dhj

dt=

1

τhj(V )

(h∞j(V ) − hj)

m∞j(V ) =1

1 + exp(Smj(Vmj

− V ))h∞j(V ) =

1

1 + exp(Shj(Vhj

− V ))

j = Na, P , K1, K2, Ca, h, hj = 1 for j = P , K2, H


Conductance-based designs (4/5)Circuit implementation by Simoni et al.

Emulation block for single ionic channel dynamics :

All circuit blocks except VOTA operate in subthreshold regionof MOSFETs ⇒ Low power consumption.


Conductance-based designs (5/5)Circuit implementation by Simoni el al.

Behavior to sustained current or no stimulus.similar to the leach heart interneurons

A. tonic firingIinj = 0pA, Eleak = 3.1V ,

gleak = 0.634nS.

B. burst firingIinj = -200pA, Eleak =

3.1V , gleak = 0.634nS.

C. burst firingIinj = 0pA, Eleak = 3.0V ,

gleak = 1.19nS.



Properties and models of biological neurons

Properties of MOSFETs

Conventional design principles for silicon neuron

Mathematical-model-based designIntroductory summaryActual designationClass 2 silicon nerve membraneClass 1 silicon nerve membraneAnother type of silicon nerve membrane


Mathematical-model-based design (1/5)This new design principle reproduces mathematical

structures of biological models.T. Kohno and K. Aihara, IEEE TNN, May, 2005

Topological structures in phase portraits andbifurcation structures explain the crucial properties ofbiological nerve membranes well.e.g. Action potential, Threshold, Refractoriness, Class 1 and 2,etc.

m

Abstraction of biological nerve membranes.


Mathematical-model-based design (2/5)Concept

Reproduce the phase portrait structures of biologicalneurons.⇒ Silicon neurons can be constructed with

“silicon-native” functions.⇓

Biologically realistic AND simple silicon neurons.




“silicon-native” functions.

⇓





“silicon-native” functions.⇓



Mathematical-model-based design (3/5)Scope

Construct an universal and elemental circuitry forsilicon neurons.

Silicon nerve membrane (SNM) circuitry based onspace clamped biological models.

Functions either as a part of or whole of neuron,according to granularity required by applications.

Adopt MOSFETs for the basic element of circuitry.


Mathematical-model-based design (3/5)Scope

Construct an universal and elemental circuitry forsilicon neurons.

Silicon nerve membrane (SNM) circuitry based onspace clamped biological models.

Functions either as a part of or whole of neuron,according to granularity required by applications.

Adopt MOSFETs for the basic element of circuitry.CCA2005 Satellite Seminar 2005.08 – p.49/69

Mathematical-model-based design (4/5)Method

1. Reproduce phase portrait structure of biologicalnerve membrane models,

focusing on topology of equilibria and nullclines,employing “MOSFET-native” curves.

Note that topology of equilibria and null-clines can not determinate everything.

e.g.existence and geometry of limit cycles.

2. Assay the bifurcation structures with computationalmethods.

Examine the silicon nerve membrane inherits thecritical properties of biological neurons.


Mathematical-model-based design (5/5)Benefits

Biologically realistic behaviors:All of the properties of biological neurons sup-ported by mathematical analyses are inherited.

Simple implementation.⇒ Suitable for large scale networks and detailed neurons.

Dynamics of other ionic channels can be addedeasily:

Relations between the ionic channels are pre-served.

Parameter tuning is easy:We know “how it works”.


Actual designation (1/4)Characteristics of MOSFETs again

Grounded source characteristics: [Quadratic curve]

Id = β2(Vi − θ)2



Differential pair characteristics: [Sigmoidal curve]

Id1 = Icmn

2+ β

4Vi

√

4Icmn

β− V 2

i

when V 2i ≤ 2Icmn

ββ : transconductance coefficient

Icmn : constant bias current


Actual designation (2/4)System equations

Conform to generalized Hodgkin-Huxley formalism.

Cy

dy

dt= − Iion(y, w1, · · · , wn),

dwi

dt=

wi,∞ − wi

Ti

Adopt two conductance parameters m and n.

Cy

dy

dt= −

y

Ry

+βm

2m2 −

βn

2n2 + a + Istim,

dm

dt=

fm(y) − m

Tm

,dn

dt=

fn(y) − n

Tn

.

As is in the H-H and M-L equations, Tm � Tn.

⇒ Cy

dy

dt= −

1

Ry

+βm

2f2

m(y) −βn

2n2 + a + Istim


Actual designation (3/4)Phase plane examples

n-V reduced H-H eqs.m → m∞, h = 0.9 − 1.2n (Rinzel, 1985)

our n-V reduced SNMat a Class 2 operation setting

Stable equilibrium (S) represents the rest state.

The ascending limb of the V(y)-nullcline gives the threshold.


Actual designation (4/4)Circuitry

+5V +5V +5V

0

mc

m1 m2

Icm

2SK213

2SK213 2SK213

-5V

+5V

-5V+5V

-5V

Vcm

V2m

+5V

n1 n2

Icn

2SJ76

2SJ76 2SJ76

+5V

-5V

+5V

-5V

Vcn

V2n

+5Vnc

-5V

-5V -5V

0

-5V

Va+

Ia+

2SJ76+5V

+5Va+

0

a-

Ia-

2SK213

-5V-5V

Va-

0

Istim

2SJ76

+5V

stim

0

Ry

Stimulus Input

R1n

R2n

R2m

R1m Cm

Cn

no

mo

n

m Vstim

0

Cy

m-blockdmdt

= fm(y)−m

Tm

n-blockdndt

= fn(y)−n

Tn


Class 2 silicon nerve membrane (1/4)Class 2 parameter setting

A subcritical Hopf bifurcation emerges in our system.

The y-nullcline moves up as thestimulus increases.⇒ The rest state loses stability.

This stable point prohibits thesystem from oscillating.⇒ The maximal point of they-nullcline should be lower.

The lower maximal point derives the higher thresholdand the lower magnitude of action potentials.



A subcritical Hopf bifurcation emerges in our system.

The y-nullcline moves up as thestimulus increases.⇒ The rest state loses stability.

This stable point prohibits thesystem from oscillating.⇒ The maximal point of they-nullcline should be lower.

The lower maximal point derives the higher thresholdand the lower magnitude of action potentials.


Class 2 silicon nerve membrane (2/4)Introduction of the forth variable h

Cy

dy

dt= −

1

Ry

+ min(βm

2fm(y)

2,βh

2h2)

−βn

2n2 + a + Istim,

dh

dt=

fh(y) − h

Th

,

dn

dt=

fn(y) − h

Tn

.

The min function voids (T) and the righter (S).⇒ Class 2 neural excitability is attained without

reducing action potential magnitudes strikingly.


Class 2 silicon nerve membrane (2/4)Introduction of the forth variable h

Cy

dy

dt= −

1

Ry

+ min(βm

2fm(y)2,

βh

2h2)

−βn

2n2 + a + Istim,

dh

dt=

fh(y) − h

Th

,

dn

dt=

fn(y) − h

Tn

.

The min function voids (T) and the righter (S).⇒ Class 2 neural excitability is attained without

reducing action potential magnitudes strikingly.


Class 2 silicon nerve membrane (3/4)Circuitry

h-blockdhdt

= fh(y)−h

Th

min function


Class 2 silicon nerve membrane (4/4)

Bifurcation diagram Experimental result

0

20

40

60

80

100

120

0.7 0.72 0.74 0.76 0.78 0.8 0.82

Freq

[Hz]

Vstim [V]

constant 0.70(V) - 0.82(V)

The same bifurcation structure to one for the M-L equations in thetype 2 setting.

Repetitive firing starts with non-zero frequency at the Hopfbifurcation point.



A saddle-node on invariant circle bifurcation emerges.

Istim just below the bifurcation(1.66 mA)

Istim just above the bifurcation(1.80 mA)

⇒

As the stimulus current Istim increases, the y-nullcline moves up.⇒ (S) and (T) approach each other until merge together and vanish.

The unstable manifolds of (T) alter to a limit cycle.


Class 1 silicon nerve membrane (2/2)

Bifurcation diagram Freq.-stimulus curve

The firing frequency is zero at the saddle-node bifurcation point N(Istim = 1.71 mA).

Arbitrarily low frequency is obtained by makingIstim close to 1.71 mA.


Another type of silicon nerve membrane (1/6)

If Cy is increased, they-component of the velocityis decreased.

⇓The amplitude of the limitcycle decreases.

The limit cycle at Istim = 1.71 mA does not reach thestable point (S), when Cy is larger than 12.83 µF.



If Cy is increased, they-component of the velocityis decreased.

⇓The amplitude of the limitcycle decreases.

The limit cycle at Istim = 1.71 mA does not reach thestable point (S), when Cy is larger than 12.83 µF.



Bifurcation diagram(Cy=14µF) Phase plane at Istim= 1.66 mA

At Istim= 1.66 mA, the right segment of the unstable manifold of(T) wraps around a stable limit cycle around (U).

If Istim is decreased, the limit cycle grows until it reaches (T).⇒ A saddle loop homoclinic orbit bifurcation takes place.



Another bifurcation that produces arbitrarily low frequency :Saddle loop homoclinic orbit bifurcation

a) (T): a saddle point.An unstable manifold of (T)

approaches a stable one.

b) The two manifolds aremerged.The unstable manifold becomes

a homoclinic manifold .




Bifurcation diagram(Cy=14µF) Freq.-stimulus curve

Bistability exists between Istim=1.61 and 1.71 mA

The system begins to oscillate with non-zero frequency andceases to oscillate with zero frequency.



Hysteresis plays a crucial role in burst firing:

A saddle loop homoclinic orbitbifurcation in the M-L eqs. :

Adding a slow feedbackcurrent can induce bursting.

(square wave bursting)

IK−Ca = gK−Caz(V − VK)

z =Cap

Cap + 1

dCa

dt= ε(−µICa − Ca)

(1) IK−Ca increases while V is silent.

(2),(3) IK−Ca decreases while V is firing repetitively.

IK−Ca acts as a dynamical stimulus current that oscillates slowly.



Burst silicon neuron

We can add a similar slow feed-back current.

Cc

dVc

dt= Ic −

Vc

Rc

,

Ic =βp

2p2,

IK−Ca =βq

2(rVc)

2.

Wave form example(square wave bursting).

The system begins to fire repetitively via a saddle-node bifurcationand ceases firing via a saddle loop homoclinic orbit bifurcation.

This additional currents can be implemented with 6 MOSFETs.


Summary (1/2)We introduced elemental knowledge on biophysicalproperties of and mathematical models on excitablecells.

Equilibrium and resting membrane potentials.Properties of the models of action potentials.

e.g. phase portrait and bifurcation structures

We introduced two conventional silicon neurondesign methods and their illustrations.

Phenomenological design.simple but not biologically realistic

Conductance-based design.biologically realistic but not simple



Equilibrium and resting membrane potentials.

Properties of the models of action potentials.e.g. phase portrait and bifurcation structures





Summary (2/2)We showed a new design method of silicon neuron.

Reproduce the phase portrait structures ofbiological nerve membrane models.⇒ biologically realistic characteristics.Tune parameters utilizing bifurcation theory.⇒ It allows us to tune up the parametersstrategically and effectively.

e.g. neuron classes and bursting

Silicon-friendly functions.⇒ Simple structure (consists of 15 MOSFETs).Based on generalized H-H eqs..⇒ Easy to add other currents.



Reproduce the phase portrait structures ofbiological nerve membrane models.⇒ biologically realistic characteristics.

Tune parameters utilizing bifurcation theory.⇒ It allows us to tune up the parametersstrategically and effectively.







Silicon-friendly functions.⇒ Simple structure (consists of 15 MOSFETs).

Based on generalized H-H eqs..⇒ Easy to add other currents.


Design of Neuromorphic Hardwareskohno/doc/ohpe050825.pdfProperties of biological neurons (4/7) What...

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