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,

then “prune” tree to smaller tree.

Reference: Avdonin & Bell, JMB, 2012; Avdonin & Bell, submitted, Inverse

Problems

Another Problem: xxt uuxquxq =++ )())(1(

From Mcgee and Johnston, J. Physiol., 1988; and

Traub, et al, J. Neurophys., 1991

Some Questions/Remarks

Subprojects:

1. Tree with variable edge diameters: conduction block

Let ϕ=u on 0×Ω∞ , so );,( ϕtxu is a solution on ],0( T×Ω

∞. For a class

A of admissible i.c.s, let ℜ→Ω∞

:, 21 ψψ have property that

21 );,( ψϕψ ≤⋅≤ tu in ],0[ T×Ω∞ . Then 21,ψψ will be a trap for A. );,( ϕtxu

will be blocked at γ if there is a trap 21,ψψ such that )(|||| ερψ ≤i ,

2,1=i and 0)( →ερ as +→ 0ε .

2. Distal current sources: Given stimulus )(tI , voltage boundary conditions on

1\ γΩ∂ , and sealed-end conditions, can we estimate 1u at 1γ (representing

potential at the soma)?

3. We have (weak) threshold conditions for FHN system on ),0( ∞×Ω :

wuwwufuu txxt µσ −=−+= ,)(

4. Want to show existence of a traveling wave solution on ],0( T×Ω∞ .

5. Want to explore (numerically) other properties, e.g., wave attenuation and

amplification along graph domains.

Query: How do populations persist in face of downstream biased flow? (The

“drift paradox”; Muller, 1954; Hershey, et al, 1993, …)

Suggestions: drifting populations represent a surplus that would not

otherwise contribute to local persistence, or compensatory upstream

movements (colonization cycle).

Lutscher, et al, 2010: populations must have ability to invade upstream in

order to persist.

Usual modeling: involves an advection-diffusion-reaction equation on a

finite interval (single river branch), or on the real line.

References above initiated persistence investigations on a branching

(metric graph) domain.

Population Persistence

in River Networks

(metric tree graph)

References:

Sarhad, Carlson, Anderson, JMB,

2012

Ramerez, JMB, 2012

Vasilyeva, Lutscher, BMB, 2012

(1) )(ufsuDuu xxxt +−= in ),0(\ ∞×Ω V

u = population density s = downstream advection speed (population

drift), which may differ from rate of current

due to organism behavior

)(uf = birth/death process. Here let 0,)( >= rruuf

Flux on

+

∂

∂−=⋅= jj

j

jjjj usx

uDAtqe ),( , jA = cross-sectional area of segment

Simplifying model assumptions: jallssDD jj _,, ==

x increases downstream

Continuity at vertex ν : 0,210 ≥== tuuu

Conservation at vertex ν : ),(),(),( 210 tqtqtq ννν +=

At interior vertex ν , incident sectional area difference is 210 AAAB −−=ν .

Assume

(2) 0)( 210 ≥−−= sAAAsBν for Ω∂∈ \Vν

Upstream boundary vertex condition: Ω∂∈= γγ ,0),( tq

Inhospitable river ending at downstream vertex dγ : 0),( =tu dγ

Reduction: ⇒= ),(),( 2/ txwetxu Dsx

(3) 0)4

(2

=−+= wD

srDww xxt in ),0(\ ∞×Ω V

(4) at dγγ \Ω∂∈ : 02

=

−

∂

∂= w

s

x

wDAq j ( )γ~je

(5) at dγ : 0),( =tw dγ

(6) at Ω∂∈ \Vν :

∂

∂−+

∂

∂−=

∂

∂−

x

wDw

sA

x

wDw

sA

x

wDw

sA 2

221

110

00222

Remark: On V\Ω , set 04

2

=−D

sr , ⇒∂

∂=

2

2

:x

DL Lwwt = is formally

self-adjoint. With Ω a metric tree graph with finitely many edges, L

can be shown self-adjoint with compact resolvent, so there is a

spectrum jλ , −∞→jλ ; and an ON basis jϕ of eigenfunctions. When

(2) 0)( 210 ≥−−= sAAAsBν at each Ω∂∈ \Vν

w equation satisfies a maximum principle. If ),( txu on Ω is continuous,

non-negative, then 0),( ≥txu for all 0>t .

Theorem: Suppose w satisfies (3)-(6), with (2) holding. Then, if

||4 1

2

λ<−D

sr , the population will not persist. (In particular, it will not

persist if 04

2

≤−D

sr .) If ||4 1

2

λ≥−D

sr a continuous positive initial

population will persist.

Remarks:

1. For the linear advection-diffusion-reaction equation on a finite

interval , with no flux at 0=x , homogeneous Dirichlet condition (hostile

river ending) at Lx = , it is an undergraduate exercise to show no

persistence (i.e., population decreases) if 1

24/ λ−≤− Dsr (and growth if

1

24/ λ−>− Dsr ).

2. On a single branch, with

)1( urusuDuu xxxt −+−= on 0,0 ><< tLx

( ) 0,0),(,0),0( >==+− ttLutsuDu xx

Vasilyeva & Lutscher, 2012 show there is no positive steady state if

04/2≤− Dsr (no persistence). For each sD, and 04/2

>− Dsr there

exists a ),(** sDLL = such that there is a unique positive steady state if

*LL > , and it is stable. We are pushing the Vasilyeva-Lutscher work to

the metric tree graph setting, using

))(1()( α−−= uuruuf

Which encompasses an allee-type mechanism.

3. Further out in consideration is to explore competition in a network

advection-driven environment; maybe with basic equations

−−+−=

−−+−=

)(

)1(

2

1

buvrvsvvDv

avuusuuDu

xxxt

xxxt

4. Return to a single population, but consider parameters that vary

depending on branch, and investigate different downstream river

ending boundary conditions.

From A. Rinaldo, et al, PNAS, 109(17), 2012.

L*-d.ol(a, TSctrLl r \qt t i eb*'c'ho'*4,

CL.a.eun, tqROl

WgL s a.t. s or.vr e*en s,o',n, g' ^L

agitrc,r ! 6e\A6ot.k €F4c€' r as '.r-\\4g A- rnr.ce,L.*'nr..^.Q. Jq.'ite' S* tftti'nl..."f\;^ *r,A- n*Ir"Lh^, ' n"r-

Sgi.er $^'\rs+ so\ve- trro Pl*ob\eu^'s

w ildn" \-^r- \, C\a r ^ q,*l t I ticrr it"rin-

o"\"r..r." "S .nrl ,",e.rrf,o'tt;"" 4o*l**d5

q*.4 so r,J'r cos ' [- od d'e-s 't"e-

t^rqb s +,,*e-hrar- ''ffet.,l 6ov\'g;tti/i'f,1r o^l d, n" oti^^Q -],t*" + mret tt't4 ?

Ho.r.e-6c-s*L s cLr€- cxa^^.g\c-3 oS

t"";+;eA €Eo.o-=l

So"J.*o,,., s ^\ , tttb t

V,t*n frin rC *."e- nwrns o op€n

hone6c,o*,rV eQ,\\s is \+ourts *.J*cl,o\crcoeg \'l^r- co*t, le w4bocs h*,,r g

nrcugoes o\ *-dh \e1p, for dc-'-ec,\r^ry ,', LD a.ln o n s ..' H4r.- ,harnSe.

g.^ rHe-A \eq 0\ bee- A.rni*t lia- Lo.v.oe-. Frn*eJ ( e"'.oo',ota,\)

cr"rnbs sVr".nlg *t\ "n,rte. t'aide,r €*cX,ran e-ic.rl , r,t,tax.rga,s

Jr^eae- Q*en\^atst&s qns ar^ 9\, grb-J .; t\ *r,tr,l t, o Pon (n*uD. )c-s*tbe . G ee 6 tr.a-v c a. h*b;+ o,E *re,e,it

. *^ 4nea- .+ qo$ab

$^novn +l^L Sse*.te, i,n *,'a$e 6.'g4€ uA^LA &r Aanoir,t& .

4.,