Extending Dynamics to Tree Graph Domains: Two Examples from Math Biology Jon Bell, UMBC 1. Cable theory and dendritic trees (Pyramidal cell from mouse cortex; by Santiago Ramon) 2. Population persistence in advection-dominated environments (river networks) (Amazon river basin: courtesy of Hideki Takayusu)
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Extending Dynamics to Tree Graph Domains: Two Examples
from Math Biology
Jon Bell, UMBC
1. Cable theory and dendritic trees
(Pyramidal cell from mouse cortex; by Santiago Ramon)
2. Population persistence in advection-dominated environments (river
networks)
(Amazon river basin: courtesy of Hideki Takayusu)
(Purkinje cells, fluorenscent dyed;
Technology Review, Dec. 2009)
−
+∂
∂=+
∂
∂=
∂
∂∑
−
m
jion
jj
mionm
i
R
v
Evwg
t
vCvI
t
vC
x
v
R
a))((
),(2 2
2
K
Example dynamics for this talk: bistable equation
)(2
2
ufx
u
t
u+
∂
∂=
∂
∂
>
<<>
<<<
==
>∈
=
∫1
0
1
0)(
10)(
00)(
0)1()0(
1],0[
:
dssf
vforvf
vforvf
ff
AACf
H f
α
α
Neuronal Cable Theory on a Dendritic Tree
(metric tree graph)
outside
inside
membrane
0hr0
hr
ihr ihr
)(0 hxv − )(0 hxv +
)( hxvi +)( hxvi − )(xvi
)(0 xv
)(xhim)( hxhim
−
)( hxhiext − )( hxhiext +)(xhiext
Problem: (1) )(2
2
ufx
u
t
u+
∂
∂=
∂
∂ on ),0(\ TV ×Ω
(2) 0=u on 0×Ω
(3) 0),(~
=∂∑ν
ν
je
j tu for 1\ γν V∈ (Kirchhoff-Neumann)
u is continuous at ν
(4) 0)(),( 11 >=∂− tItu γ for ],0[ Tt ∈
1fH : There exists 0, 00 >fu such that for 00 uu ≤≤ , ufuf 0)( −≤ .
Let ( ) ( )],0(\],0[: 1,2TVCTCZ
T×Ω∩×Ω= .
Theorem: Let f satisfy 1, ff HH . Let T
Zu ∈ be a solution to (1)-(4), for any T
> 0, where
≥
<<=
−−
0
)(
1
00
0
0)(
ttforeI
ttforItI
ttδ . Here 010 0,0 fII <<>≥ δ . Then
0),(lim =∞→
txu jt
for all jex ∈ , all j = 1,…,N.
2fH : for some 0>bf , ufuf b−≥)( for 0≥u .
Theorem: Let f satisfy 1, ff HH . Let TZu ∈ solve (1)-(4) with )(tI replaced by
)(* tIµ , where 0)(* ≥tI , *I not identically zero. Then there exists a 0µ
depending on *I , such that if 0µµ > , for each j, ),(),( txvtxu jj ≥ and
1),(lim =∞→
txv jt
.
VE ∪=Ω
MN VeeeE ννν ,...,,,...,, 2,121 ==
mindexV γγγνν ,...,,1)(| 21==∈=Ω∂
2)(|\ >∈=Ω∂ νν indexVV
=Ω metric graph if every edge Ee j ∈ is
identified with an interval of the real
line with positive length jl .
=Ω tree graph if there are no cycles.
Bounds on the Speed of Propagation: IVP example
Consider the problem in ),0( ∞×Ω∞ :
Assume u is a solution to (IVP) such that
(*) 1lim =∞→
jt
u in ),0( ∞×Ω∞
Write )()()0(')( ugauugufuf +−=+= , g is smooth, g(u) = O(u2) as
0→u . Define 0/)(sup:10
>=<<
uugu
σ .
Note that L 0)(: ≤−=−+−= jjjjjxxjtj uuguauuuu σσ .
Assume 3,2,1,10 =≤≤ jjφ . Then (by another comparison result),
10 ≤≤ jv .
Theorem: Suppose v is a solution to (IVP) satisfying (*). Let φ have
bounded support, supp 1e⊂φ . If σσ +=> acc /2: , then for each j, each
jex ∈ , 0),(lim =+∞→
tctxu jt
.
Suppose (IVP) admits a positive steady state solution q(x) for
0<≤≤ bxa , with q(a) = q(b) = 0. Suppose the only nonnegative global
steady state τ of (IVP) with )()(1 xqx ≥τ on [a,b] is 1≡τ . Let u be a
solution to (IVP) with )()(1 xqx ≥φ on [a,b] .
Then there is a 0>c such that for cc <<0 , for any x, any 0>ε , there
is a T > 0 such that for t > T, 3,2,1,1),( =−≥+ jtctxu j ε .
(IVP) =
>+
×Ω=
×Ω+=
0:
0
),0(\)(
tatcontinuityKN
inu
TVinufuu xxt
γ
φ
Inverse Problem
Motivation:
Problem: (1) xxt uuxqu =+ )( on ),0(\ TV ×Ω
(2) 0=u on 0×Ω
(3) 0),(~
=∂∑ν
ν
je
j tu for Ω∂∈ \Vν
u is continuous at Ω∂∈ \Vν
(4) )],,0([2 m
TLfu ℜ∈=∂ on ],0[ T×Ω∂
Strategy: determine q(x) on boundary edges via Boundary Control Theory,