(a)Bistability: returntoequilibriumviathesame
im-pulse
0 1000 2000 3000 4000 5000 6000
−5 0 5 10
0 1000 2000 3000 4000 5000 6000
−5 0 5 10
(P) bistability
(b)Bistability
Figure9: Bistabilityphenomenon: Therst impulseindues aself-sustainedtoni spiking
behaviorwhilethesystemhasastablexedpoint. Theseondimpulseperturbsthisregular
spikingbehaviorandthesystemfalls intheattrationbasin ofthestablexed point.
(xviii)/(xx) Self-sustainedsubthreshold osillations and purely osillating mode: they
arelinkedwiththesuperritialHopfbifurationanditsstableperiodiorbit. These
twobehaviors annot be obtainedin theIBG models sinethe Hopfbifuration are
alwayssubritial.
4.3 Self-sustained subthreshold osillations in ortialneurons
Inthis studywegaveasetof suientonditionsto obtainanIBG-likemodel ofneuron.
InthisframeworkweproposedamodelthatdisplaysaBautinbifurationtheIBGneurons
lak; as a onsequene our model an produe subthreshold osillations. In this setion,
weexplainform abiologialpointof viewtheoriginandtheroleof thoseosillations,and
reproduein vivoreordings.
In theIBG models, theAndronov-Hopf bifuration is alwayssubritial. Theonly
os-illationsreatedinthese modelsaredamped(seeFig 11(a)),andorrespondin thephase
planetotheonvergenetoaxedpointwheretheJaobianmatrixhasomplexeigenvalues.
OurquartimodelundergoessuperritialAndronov-Hopfbifurations,sothereare
attrat-ing periodi solutions. This meansthat theneurons anshowself-sustained subthreshold
osillations(Figs. 11(b)and11())whihisofpartiularimportanein neurosiene.
Mostbiologialneuronsshowasharptransitionfromsilenetoaspikingbehavior,whih
isreproduedinallthemodelsoflass1.1. However,experimentalstudiessuggestthatsome
neuronsmay experienearegimeofsmall osillations[22℄. Thesesubthresholdosillations
anfailitate the generation of spike osillations when the membrane gets depolarized or
(i) Tonic Spiking (ii) Phasic Spiking (iii) tonic bursting
(iv) phasic bursting (v) Mixed mode (vi) Spike freq. adaptation
(vii) Class 1 excitability (viii) Class 2 excitability (ix) Spike latency
(x) Damped subthr. oscill. (xi) resonator (xii) integrator
(xiii) rebound spike (xiv) rebound burst (xv) Threshold variability
(xvi) bistability (xvii) depol. after−pot. (xviii) self−sustained oscill.
(xix) Mixed chatter/C1 exc. (xx) Purely Oscill. mode
Figure10: PhasediagramsorrespondingtothebehaviorspresentedinFig 5.
0 5 10 15 20 25 30 35 40 45 50
−2
−1 0 1
0 5 10 15 20 25 30 35 40 45 50
−2 0 2
−1.5 −1 −0.5 0 0.5 1
−2 0 2
(a) Dampedosillations
0 2000 4000 6000 8000 10000 12000 14000 16000
−0.635
−0.63
−0.625
−0.62
0 2000 4000 6000 8000 10000 12000 14000 16000
−1.58
−1.575
−1.57
−1.565
−0.633 −0.632 −0.631 −0.63 −0.629 −0.628 −0.627 −0.626 −0.625 −0.624
−1.58
−1.575
−1.57
−1.565
(b)Transientphasetowardsthe
self-sustainedosillations
0 2000 4000 6000 8000 10000 12000
−1
−0.5 0
0 2000 4000 6000 8000 10000 12000
−2
−1.5
−1
−0.5
−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1
−2
−1.5
−1
−0.5
() Self-sustained osillations
(stationnarystate)
Figure11: ThequartimodelshowsdampedsubthresholdosillationsliketheIBGmodels
(Fig. 11(a)): the trajetory ollapses to axed point(
a = 1, b = 1.5, I = 0.1, T max = 100, dt = 0.01
. Theupper(blue)urverepresentsthesolutioninv
,themiddle(red)onew
andthelast onethe trajetoryintheplane
(v, w)
. Self-sustained subthresholdosillations ofthequartimodel(Figs11(b)and11()): thetrajetoryisattratedtowardsalimityle(parameters:
a = 1, b = 5/2, I = − 3(a/4) 4/3 (2a − 1), T max = 150000, dt = 0.01, I = ( − 3(a/4) 4/3 (2a − 1) + 0.001)
hyperpolarized [23, 24℄. They also play an important role in shaping spei forms of
rhythmiativitythat arevulnerabletothenoiseinthenetwork dynamis.
Forinstane, theinferiorolivenuleus, apartofthe brainthat sendssensory
informa-tiontotheerebellum,isomposed ofneuronsableto supportosillationsaroundtherest
potential. IthasbeenshownbyLlinásandYarom[23,24℄thatthepreisionandrobustness
ofthese osillationsareimportant forthepreisionand therobustnessofspikegeneration
patterns. The quarti model is able to reprodue the main features of the inferior olive
neurondynamis:
i. autonomous subthreshold periodi andregular osillations. (see intraellular
reord-ingsofinferioroliveneuronsinbrainstemsliesin [24℄).
ii. Rhythmigenerationofationpotentials.
Therobustsubthreshold osillationsshownbyin vivoreordings[4,21, 24℄ orrespond
inourquartimodeltothestablelimityleomingfromthesuperritialHopfbifuration.
Theosillationsgeneratedbythisyleare stable,and theyhaveadeniteamplitudeand
frequeny. This osillation ours at thesametime that the rhythmi spike generation in
presene of noisyor varying input. Note that other neuron models suh as those studied
above,eveniftheydonotundergoasuperritialHopfbifuration,analsoexhibit
osilla-tionsinthepreseneofnoise,forinstanenearasubritialHopfbifuration. Nevertheless,
theseosillationshavenottheregularityintheamplitudeandthefrequenylinkedwiththe
presene of anattrating limit yle. The resultswe obtainsimulatingthe quarti model
areverysimilartothoseobtainedbyin vivoreordings(seeg. 12).
Buttheinferioroliveneuronsarenottheonlyneuronstopresentsubthresholdmembrane
potentialosillations. Forinstane,stellateellsintheenthorhinalortexdemonstratetheta
frequenysubthreshold osillations[1,2,17℄),linkedwiththepersistantNa
+
urrent
I
NaP.Wenowonludethissetiononthespeiexampleofsubthresholdself-sustained
osil-lationsgivenbythedorsal rootganglia(DRG) neuron. This neuronpresentssubthreshold
membranepotentialosillationsoupledwithrepetitivespikedishargeorburst,forinstane
in theaseofanerveinjury[20, 3℄. The gureFig.12(d), arebiologialin vivo
intraellu-larreordingsperformed byLiu et al [20℄ from aDRGneuron of an adultmale rat. The
reordedmembranepotentialexhibithighfrequenysubthresholdosillationinthepresene
ofnoise,ombinedwitharepetitivespikingorbursting. Thesebehaviorsanbereprodued
bythe quartimodelasweansee in the gureFig.12, around apointwhere thesystem
undergoesasuperritialHopfbifuration 7
.
Conlusion
Inthis paperwedened a generallass of neuronmodels ableto reproduea widerange
of neuronal behaviors observed in experiments on ortial neurons. This lass inludes
the Izhikevih and the Brette-Gerstner models, whih are widely used. We derived the
bifurationdiagramoftheneuronsofthislass,andprovedthattheyallundergothesame
typesofbifurations: asaddle-nodebifurationurve,anAndronov-Hopfbifurationurve
andaodimension2Bogdanov-Takensbifuration. Weprovedthattherewasonlyoneother
possiblexed-pointbifuration, aBautinbifuration. Then using thosetheoretial results
we proved that the Izhikevih and the Brette-Gerstner models had the same bifuration
diagram.
Thistheoretialstudyallowsustosearhforinterestingmodelsinthislassofneurons.
Indeed,theorem1.5ensuresusthatthebifurationdiagramwill presentatleastthe
bifur-ationsstated. This informationis ofgreatinterestif wewanttoontrol thesubthreshold
behavioroftheneuronof interest.
Followingthese ideas, we introdued anew neuronmodel of ourglobal lass
undergo-ing the Bautinbifuration. This model, alled the quarti model, is omputationally and
mathematiallyas simpleastheIBG models, and ableto reproduesomeortial neuron
behaviorswhihtheIBGmodelsannotreprodue.
Thisstudyfousedonthesubthresholdpropertiesofthislassofneurons. Theadaptative
reset of the model is of great interest and is a key parameter in the repetitive spiking
propertiesoftheneuron. Itsmathematialstudyis veryrih,andisstillanongoingwork.
7
Theamplitudeandfrequenyofthesubthresholdosillationsanbeontrolledhosingapointonthe
superritialHopfbifurationurve.
0 0.5 1 1.5 2 2.5 3 x 104
−2
−1 0 1 2 3 4 5
(a)Withoutspiking
3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
x 104
−2
−1.5
−1
−0.5 0 0.5 1 1.5 2 2.5 3
(b)Withintermittentspiking
2.5 3 3.5 4 4.5 5 5.5 6
x 104
−2
−1.5
−1
−0.5 0 0.5 1 1.5 2
() Withintermittentbursting
(d)Biologialreordings
Figure 12: Subthreshold membrane osillations, qualitatively reproduing the reordings
from[20℄indorsalrootganglion(DRG)neurons. Traesillustrate(12(a))osillations
with-outspiking, (12(b)) osillationswith intermittent spiking and (12())osillations with
in-termittentbursting. (in thegures,spikesaretrunated). Thenoisyinputis an
Ornstein-Ulhenbek proess. The biologial reordings 12(d) are reprodued from [20, Fig.1℄ with
permission.
Aknowledgments
The author warmly aknowledges T. Viéville, B. Cessa,Olivier Faugerasfor fruitful
dis-ussionsandsuggestions.
A Bautin bifuration
In this appendix we prove that the quarti model undergoesa Bautin bifuration at the
point
b = 5 2 a I = − 3 a 4 4/3
(2 a − 1) v a = − a 4 1/3
(A.1)
A.1 The rst Lyapunov exponent
Indeed,usingasuitableanehangeofoordinates,thesystematthispointreads:
˙ x = ωy
˙
y = ab ω 6v 2 a v 1 (x, y) 2 + 4v a v 1 (x, y) 3 + v 1 (x, y) 4
= 1 2 F 2 ( x y , x y
) + 1 6 F 3 ( x y , x y
, x y
) + 24 1 F 3 ( x y , x y
, x y , x y
)
(A.2)
where
v 1 (x, y) = 1 b x + ab ω y
. We also denoteF 2 (X, Y )
,F 3 (X, Y, Z)
andF 4 (X, Y, Z, T )
themultilinearsymmetrivetorfuntions of (A.2)(
X, Y, Z, T ∈
R2
).( F 2 ( x y , z t
) = 12 ab 0
ω v 2 a v 1 (x,y)v 1 (z,t)
. . .
To ompute the two rst Lyapunov exponents of the system, we follow Kuznetsov's
method [19℄. In this method we need to ompute some spei right and left omplex
eigenvetors,whihanbehoseninouraseto be:
p =
1
− i √ a b − a 2 +a
1
!
q =
1 2
(i √
a(b − a)+a)b b − a − i √
a(b − a)
1/2 (i
√ a(b − a)+a) 2 a (b − a − i √
a(b − a))
(A.3)
Wenowputthesysteminaomplexformletting
z = x + iy
g ij
g 20 =< p, F 2 (q, q) >
g 11 =< p, F 2 (q, q) ¯ >
g 02 =< p, F 2 (¯ q, q) ¯ >
g 30 =< p, F 3 (q, q, q) >
g 21 =< p, F 3 (q, q, q) ¯ >
g 12 =< p, F 3 (¯ q, q, ¯ q) ¯ >
g 03 =< p, F 3 (¯ q, q, ¯ q) ¯ >
. . .
(A.4)
SotheTayloroeients(A.4) read:
g 20 = 12 ab ω v 2 a v 1
1 2
(i √
a(b − a)+a)b b − a − i √
a(b − a) , 1 2 (i
√ a(b − a)+a) 2 a (b − a − i √
a(b − a))
2
g 11 = 12 ab ω v 2 a v 1 (q)v 1 (¯ q) g 02 = 12 ab ω v 2 a v 1 (¯ q)v 1 (¯ q) . . .
(A.5)
Letnow
S(I, b) := F ′ (v − (I, b))
bethevalueof thederivativeofthefuntionF
, denedaroundthebifurationpointweareinterestedin.
TheJaobianmatrixintheneighboorhoodofthepoint(A.1)reads:
L(v) =
S(I, b) 1 ab − a
Letusdenote
α = I b
theparametervetor,
λ(α) = µ(α) ± iω(α)
theeigenvaluesofthe Jaobianmatrix. Wehave:( µ(α) = 1 2 (S(α) − a) ω(α) = 1 2 p
− (S(α) − a) 2 + 4ab
With thesenotations,let
c 1 (α)
betheomplexdened by:c 1 (α) = g 20 g 11 (2λ + ¯ λ)
2 | λ | 2 + | g 11 | 2
λ + | g 02 | 2
2(2λ − ¯ λ) + g 21
2 .
(inthisformulaweomitthedependane in
α
ofλ
forthesakeoflarity.TherstLyapunovexponent
l 1 (α)
eventuallyreads:l 1 (α) = Re(c 1 (α))
ω(α) − µ(α)
ω(α) 2 Im(c 1 (α))
(A.6)A.2 The seond Lyapunov exponent
ThemethodtoomputetheseondLyapunovexponentisthesameastheonewedesribed
intheprevioussetion. Theexpressionisgivenbythefollowingformula:
2l 2 (0) = 1
ω(0) Re[g 32 ]
+ 1
ω(0) 2 Im[g 20 g ¯ 31 − g 11 (4 g 31 + 3 ¯ g 22 ) − 1
3 g 02 (g 40 + ¯ g 13 ) − g 30 g 12 ]
+ 1
ω(0) 3 { Re[g 20
¯
g 11 (3 g 12 − g ¯ 30 ) + g 02 ( ¯ g 12 − 1/3 g 30 ) + 1 3 g ¯ 02 g 03
+ g 11 ( ¯ g 02
5
3 g ¯ 30 + 3 g 12
+ 1
3 g 02 g ¯ 03 − 4 g 11 g 30 )]
+ 3 Im[g 20 g 11 ]Im[g 21 ] }
+ 1
ω(0) 4 { Im[g 11 g ¯ 02 g ¯ 20 2
− 3 ¯ g 20 g 11 − 4 g 11 2 ] + Im[g 20 g 11 ] 3 Re(g 20 g 11 ) − 2 | g 02 | 2
}
This expression is quite intriate in ourase. Nevertheless we havea losed-form
ex-pressiondepending on theparameter
a
, vanishing for twovalues of theparametera
. WeevaluatenumeriallythisseondLyapunovexponent. Wegetthefollowingexpression:
l 2 (a) ≈ − 0.003165 a − 28 3 − 0.1898 a − 22 3 + 0.3194 a − 16/3
− 0.05392 a − 25 3 + 0.1400 a − 19 3 − 0.3880 a − 7/3 + 0.5530 a − 10/3 +0.7450 a − 13/3 .
(A.7)
Weansee thatthis numerialexponentvanishesonlyfor twovaluesof theparameter
a
whihare{ 0.5304, 2.385 } .
The expression of the determinant of the matrix
D I,b (µ(I, b), l 1 (I, b))
are even moreinvolved,sowedonot reprodue ithere(it would takepagesto write down its numerial
expression!). Nevertheless,weproeedexatlyaswedidfortheseondLyapunovexponent
andobtainagaintherigorousresultthatthis determinantnevervanishesforall
a > 0
.InthisannexwegivethenumerialvaluesusedtogenerateFig. 5.
(i)ToniSpiking (ii)PhasiSpiking (iii)ToniBursting
a = 1
;b = 0.49
;v r = 0
;a = 1
;b = 0.76
;v r = 0.2
;a = 0.15
;b = 1.68
;v r = ( − 2a + b) 1 3
;I(t) = 1.56
1t>1 (t)
;d = 1
;I = 0.37
1t>1 (t)
;d = 1
;I = 4.67
1t>1 (t)
;d = 1
;T = 10
;dt = 0.01
;θ = 10
;T = 10
;dt = 0.01
;θ = 10
;T = 30
;dt = 0.01
;θ = 10
;(iv)PhasiBursting (v)MixedMode (vi)SpikeFreq.Adaptation
a=1.58;b=1.70;
v r = − a 4 1
3
; a=0.07;b=0.32;v r =
0; a=0.02;b=0.74;v r =
0;I(t) =0.73
1t>1 (t)
;d =
0.01;I(t) =3.84
1t>1 (t)
;d =
1.50;I(t) =4.33
1t>1 (t)
;d =
0.36;T =
50;dt =
0.01;θ =
10.T =
50;dt =
0.01;θ =
10.T =
50;dt =
0.01;θ =
10.(vii)Class1Exitability (viii)Class2Exitability (ix)SpikeLateny
a=4;b=0.67;
v r =
-1.3; a=1;b=1.09;v r =
-1.2; a=0.02;b=0.42;v r =
0;I(t) = − 0.1 + 0.23t
;d =
1;I(t) =0.06t
;d =
5;I(t) =5δ 7.5 (t)
;d =
1;T =
30;dt =
0.01;θ =
10.T =
50;dt =
0.01;θ =
20.T =
15;dt =
0.01;θ =
10.(x)DampedSubthr.Osill. (xi)Resonator (xii)Integrator
a=2.58;b=4.16;
v r =
0.1; a=5.00;b=7.88;v r =
-1.28; a=1.00;b=1.10;v r =
-0.97;I(t) =2δ 2 (t)
;d =
0.05;I(t) =δ 6,6.8,15,16.5,24,26 (t)
;d =
0.5;I(t) =δ 2.5,3.3,17.5,19 (t)
;d =
0.5;T =
20;dt =
0.01;θ =
10.T =
30;dt =
0.01;θ =
10.T =
25;dt =
0.01;θ =
10.(xiii)ReboundSpike (xiv)ReboundBurst (xv)Thresholdvariability
a=1;b=2;
v r =
-0.63; a=1;b=2;v r =
1.3; a=1;b=1.23;v r =
-0.91;I(t) = − 0.48 − 5δ 2.5 (t)
;d =
1;I(t) = − 0.48 − 30δ 6.5 (t)
;d =
1;I(t) =δ 2,16.5 − δ 15
;d =
1;T =
50;dt =
0.1;θ =
10.T =
20;dt =
0.01;θ =
10.T =
20;dt =
0.01;θ =
10.(xvi)Bistability (xvii)Depol.after-pot (xviii)Self-sustainedosill.
a=1;b=1.2;
v r =
0.8; a=1;b=1.5;v r =
0.06; a=1;b=2.5;v r =
-0.63;I (t) = − 0.47 + 20 ∗ (δ 10 − δ 30 )
;d =
0.5;I(t) =2δ 3
;d =
0.01;I(t) = − 0.475 + 10 ∗ δ 10
;d =
1;T =
50;dt =
0.01;θ =
10.T =
30;dt =
0.01;θ =
10.T =
100;dt =
0.01;θ =
10.(xix)MixedChatter/
C 1
ex. (xx)Purelyosill.a=0.89;b=3.65;
v r =
1.12; a=1;b=2.6;v r =
-0.63;I(t) =0.07t
;d =
1;I(t) = − 0.47
1t>1
;d =
1;T =
50;dt =
0.01;θ =
10.T =
500;dt =
0.01;θ =
10.Remark. The
δ u (t)
funtionisdenedby:δ u1,...uN (t) = 8
<
:
1
ift ∈ S
k∈{1,...N}
[u k , u k + 0.3]
0
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