Neuronal Cable Theory (NCT) on Graphs: A Synopsis of Some Forward and Inverse Problems

by Jonathan Bell

Outline

Motivation: Some comments on dendritic trees and neural transmission NCT forward problems: aspects of threshold behavior, bounds on propagation speed, and conduction block NCT inverse problems: somatic response to distal input, ion channel densities, conductances and branch radii, and determining tree morphology from boundary measurements

Dendritic Tree Examples

Purkinje cell with red fluorescent dye injected into it (and another cell with green dye. (From Technology Review, Dec. 2009)

Other Branching Systems

Comments on Dendrites Dendrites account for more than 99% of surface area of some neurons, and can be studded with up to 200,000 synaptic inputs They are the largest volumetric component of nervous tissue and consume more than 60% of the brain’s energy They are computing workhorses of the brain A bit of history of ideas about dendrites: • Before electrophysiology, dendrites were considered

to play a strictly nutritive role to neurons (like tree roots)

• The “classical” picture emerged in the 1930s and 40s which held that the exclusive spike trigger zone in nerve was the axon initial segment, while dendrites simply collected and summated synaptic inputs

• Now many dendrites are known to be replete with complex voltage-dependent membrane conductances capable of action potential production and other highly nonlinear behaviors.

Cable Equation(s)

Single branch circuit model ⇒ continuum model:

⎪⎩

⎪⎨

⎧ −

+∂∂

=+∂∂

=∂∂

∑−

m

jionjj

mionmi

Rv

Evwg

tvCvI

tvC

xv

Ra

))((

),(2 2

2

K

outside

inside

membran

0hr 0hr

ihr ihr

)(0 hxv − )(0 hxv +

)( hxvi +)( hxvi − )(xvi

)(0 xv

)(xhim)( hxhim −

)( hxhiext − )( hxhiext +)(xhiext

Idealized Dendritic Tree Graph (graph = tree graph = graph with no cycles = network)

1. Trees are planar, binary; if one terminal vertex identified as a soma, then 1|| +=Ω∂ n

2. If one can go from the soma to any one terminal node traversing N branch points, the tree is complete and has order N. The tree example above is incomplete, but has “bounded order” Nb= 4. Tree complete

Nn 2=⇒ . 3. Most of time, graph = compact, metric tree graph { }VE,=Ω { }NjeE j ,,1| K== , ],0[ jj le ≅ { } ri VMiV ∪Ω∂=== ,,1| Kγ

soma

Dendrogram of a CA1 Pyramidal Cell

In this example, n = 95, Nb = 33 .

In this example, n = 90, Nb = 11 .

Some Notation and Example System

Example Problem: FitzHugh-Nagumo

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

>

<<><<<

==>∈

=

∫1

0

1

0)(

10)(00)(

0)1()0(1],0[

:

dssf

vforvfvforvf

ffAACf

H fα

α

Dynamics:

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

>=Ω

−=∂

∂

−+∂

∂=

∂

∂

0,,,,1,

)(2

2

ησ

ησ

Njin

wvt

w

wvfxv

tv

jT

jjj

jjjj

K

IC: 00,0 jjj onwv Ω==

BC: 1,~},{\,0),(],0()(),(

0

01

>Ω∂∈=∈=−

jetvTttItv

jjx

x

γγγγγ

(sealed-end b.c.s)

K (Kirchhoff condition): ],0(,0),(,

~,TttvdV

ijej ijxijri ∈=∈ ∑ γγγ

],0(),0(:1

Tl j

N

jT ×⎟

⎠⎞

⎜⎝⎛=Ω

=U

],0(),0(: Tl jjT ×=Ω

}0{:0 ×=Ω jj e

D = incident matrix = (dij),

⎪⎩

⎪⎨

⎧=−=

=otherwise

iflif

d i

ji

ij

001

1γγ

All edges have the same “radius”

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−

−

=

00100100000100011010

0011100001

D

Threshold Idea: Motivation through Energy Arguments Example: let ),1)(()( vvvvf −−= α 2/10 <<α , so

∫ ⎥⎦

⎤⎢⎣

⎡+

+−−==

vvvvdssfvF

0

22

431

4)(:)( αα

So 2121

1

1,0)(,00)(

aaavaforvFavforvF

<<<<<><<<

α

Let

∑ ∫=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+++++−⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂=

N

j

l

jjjjjjjj

j

dxwvKwvkvdvkFxvktE

10

22222

1 )(2

)(2

)(2

)(2

:)(

Lemma: There are k, d, K such that if, for some 00 ≥t , every j = 1, …, N,

,)(,|),(|sup 010 αα <<∈

tEtxv jex j

then ( ) ( )0,0),(),,(lim =∞→

txwtxv jjt.

Let

221

0

2

2 21

21:),(,),()(

21:)( CwBvwAvwvBLdxwvBLvF

xv

tE N

j

l

jjjjj

j

+−=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+−⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂= ∑ ∫=

By a proper choice of A, B, C, 0),( ≥jj wvBL , for j = 1, …, N, and for solution (v,w), 0/2 ≤dtdE . Then, Lemma: Assume 02 >≥ σν , and for some 00 ≥t , 0)( 02 <tE . Then

0)(2 <tE for all 0tt ≥ and this implies for every 0tt ≥ , there is a },,1{ Nk K∈ , )(txx k= such that α>> 1)),(( attxv kk .

Bottom Line: (0,0) is locally stable, but not globally stable.

Threshold Behavior: Scalar Dynamics Case

FitzHugh-Nagumo model, but with 0≡jw , plus assume

1fH : There exists 0, 00 >fu such that for 00 uu ≤≤ , ufuf 0)( −≤ . From a comparison principle… Theorem: f satisfies 1, ff HH ; let )(]),0[(: 1,2

TT CTCZu Ω∩×Ω=∈ be a solution, for any T > 0, (1) jTjjxxjt inufuu Ω+= )(

(2) ⎩⎨⎧

≥<<

==− −−0

)(1

000 0

0)(),(

ttforeIttforI

tItu ttix δγ

(3) }{\0),( 0γγγ Ω∂∈= fortu jx , j is such that γ~je (4) rVfortuK ∈= γγ 0);,( (5) 00 jj inu Ω= .

Here 010 0,0 fII <<>≥ δ . Then 0),(lim =∞→

txu jt for all jex∈ , all j

= 1,…,N. Assume

2fH : for some 0>bf , ufuf b−≥)( for 0≥u . Theorem: Let TZu∈ solve (1)-(5) with I(t) 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

. Here f satisfies 2, ff HH .

Bounds on the Speed of Propagation: IVP example

Consider the problem in ),0( ∞×Ω∞ :

Assume v is a solution to (IVP) such that (6) 1lim =

∞→ jtv in ),0( ∞×Ω∞

Write )()()0(')( ugauugufuf +−=+= , g is smooth and g(u) = O(u2) as 0→u . Define { } 0/)(sup:

10>=

<<uug

uσ .

Note that L 0)(: ≤−=−+−= jjjjjxxjtj vvgvavvvv σσ . Assume 3,2,1,10 =≤≤ jjφ . Then (by another comparison result),

10 ≤≤ jv . Theorem: Suppose v is a solution to (IVP) satisfying (6). Let φ have bounded support, supp 1e⊂φ . If σσ +=> acc /2: , then for each j, each

jex∈ , 0),(lim =+∞→

tctxv 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 v 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),( =−≥+ jtctxv j ε .

(IVP) =

⎪⎩

⎪⎨

⎧

>=Ω=Ω+=

00);,(

)(

0

tfortvKinvinvfvv

jjj

jTjjxxjt

γφ

Conduction Block

It is slightly more convenient to rescale; let ⎩⎨⎧

=32

23

1

,/ eeinaxeinx

z . Then

)( jjxxjjt vfvDv += , where ⎪⎩

⎪⎨

⎧=

3

2

11:

eineinein

D j

εβε . Here 32 / aa=β and

33−= aε . Also ( ) 0),( ),|(32

21 =−−= tzzz vvvtvK γβ .

Definition: For a class A of admissible initial functions, let ℜ→Ω∞:, 21 ψψ have the property ∈φ A implies jjj tv 21 );,( ψφψ ≤⋅≤ in 3,2,1, =Ω jjT . Then { }21,ψψ will be called a trap for A. Definition: );,( φtxv will be said to be blocked at γ if there is a trap { }21,ψψ such that asandii 0)(,2,1),(|||| →=≤+ℜ

ερερψ +→ 0ε . Remark: This requires a (conditional) comparison principle to help construct such a trap. We have an approach to do this for bistable dynamics above, and believe it doable for FitzHugh-Nagumo (as a generalization of a result of Pauwelussen, 1981) and Morris-Lecar (as a generalization of a result of Zhou-Bell, 1994).

Return to (IVP) on ∞Ω that includes “pseudo-uniform” radii:

⎪⎩

⎪⎨

⎧

>=Ω=Ω+=

00),(

)(

0

ttvKinvinvfvav

jjj

jTjjxxjjt

φ

where, for convenience, 11 =a , and ),0(),0(),0(:),( 3

232

221 tvatvatvtvK xxx +−+−−=

Somatic Output due to a Distal Current Source

Present Cases

A. Linear cable theory: jTsjkj

jjj inxxtRI

xv

avt

vΩ−+

∂

∂=+

∂

∂)()(2

2

δδ

plus zero i.c.s, sealed-end b.c.s and voltage measurements on { }0\ γΩ∂ , continuity and Kirchhoff conditions at on Vr , continuity and jump conditions at x = xs :

[ ] ,0),(),(:|| =−−+= −+ txvtxvv skskxk s [ ] kxkx atRIv

s/)(|| −=∂

Key is use of Fourier transform in t to obtain ∫∞

∞−

= ωωπ

ω daetv it )(21),0( 11 .

Methodology works in principle for a finite number of point stimuli.

B. Nonlinear cable theory: ⎪⎪⎩

⎪⎪⎨

⎧

−=∂

∂

−+−+∂

∂=

∂

∂

jjj

sjkjjj

jj

wvt

w

xxtRIwvfxv

at

v

βα

δδ )()()(2

2

Only can address question for steady state case, βα ,),(vf such that βα /)( vvf − satisfies hypothesis fH .

Key here is uniqueness of trajectories through non-singular points in phase plane, and knowledge of lengths of trajectory segments (length of edges) Problem: can not do anything with the (ill-posed) diffusion problems.

Question: Given stimulus I(t), voltages mj(t) and sealed-end b.c.s on { }0\ γΩ∂ , can we estimate v1(0,t) at

0γ (representing potential at the soma)?

Inverse Problem for a Dendritic Tree Example: Measurements of diameters (upper numbers) and segment lengths (lower numbers) of a primary dendrite and its branches from a cat spinal motoneuron. (Lengths are not to scale.) (data from J. Barrett, as presented by H. Tuckwell in Introduction to Theoretical Neurobiology, vol. 1, Cambridge Univ. Press, 1988.)

Question Given sufficient electrical data at the terminal ends of a dendritic tree, can we “recover the tree” in the sense of obtaining the topology of the tree (up to isometry), the lengths of the branches (edges), the diameters of the branches, and the conductances in the branches? Example: smallest case of two non-isometric trees (one complete, one incomplete)

A Single Branch Example: Recover a Channel Density

Assumptions: single (unbranched) passive cable single ion channel with distributed density N(x) *)(*,)(0 gxNgCxNCCm =+= C0 = capacity of membrane without channels (~0.8 2/ cmFμ ) C* = capacity per channel (~ F18108 −× ) g* = maximum conductance per single channel (~ 112105.2 −− Ω× ) Model Problem:

⎪⎪

⎩

⎪⎪

⎨

⎧

==∂∂

−=∂∂

<<=

<<<<∂∂

=+∂∂

+

)(),0(0),(),(),0(

00)0,(

00,)())(1( 2

2

tftvtlxvtit

xv

lxxv

Ttlxxvvxq

tvxq

Inverse Problem: Find a constructive theory for recovering q(x) to this problem and its graph analogue (Avdonin & B: boundary control method…work in progress, along with problem on a graph) Lemma (mentioned in B and Craciun, 2005): Let 21,vv be solutions to (6) corresponding to ],0[, 2

21 lLqq ∈ , for some stimulus i(t), where ),,0(2 TCi∈ with 0)0(',0)0( ≠= ii . If ),0(),0( 21 tvtv = on (0,T), then )()( 21 xqxq = on [0,l]. Remark: One parameter, but different, more challenging, than the classical physics case of xxt vvxqv =+ )( . Another case (B and Craciun, 2005): Morris-Lecar…recover K+ channel density

⎭⎬⎫

⎩⎨⎧ −

=∂∂

∂∂

=−+−+−+∂∂

+

∞

∞

)()(

)()()())(())(1( 2

2

vwvw

tw

xvEvgEvwxqEvvmg

tvxq llKCaCa

τϕ

Single Branch IPs (continued) Another non-traditional IP:

⎭⎬⎫

⎩⎨⎧

∂∂

∂∂

=+∂∂

xvxa

xxavxg

tv 2)(

)(1)( , 0,0 ><< tlx

Separate variables: ⇒= )()(),( xtTtxv ϕ

0)(2 =−+⎟⎠⎞

⎜⎝⎛ ϕλϕ ag

dxda

dxd

.

Liouville transformation: ,)(

10∫=x

sads

Ly ∫=

l

sadsL

0 )(: , ϕ4/3: aw =

(1) 10,0))(( 22

2

<<=−+ ywyQdy

wd μ standard form of S-L operator

where ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛++==

2

2

22222

41

43

dxda

adxadLgLQandLλμ .

Much is known about (1). For example, if w’(0)-hw(0) = 0 = w’(1), then

)/1()2cos()()( 21

0

1

0

222 nOdyynyQdyyQnn +−+= ∫∫ ππμ as ( ))1,0(, 2LQn ∈∞→

Through spectral theory there are classical (and not so classical) ways of determining Q. We assume that is done. Can not recover both a(x), g(x), but

A. Given a smooth )(xaa = , then ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+=

2

2

2

41

43)(

dxda

adxadxb is known, so

)(/)()( 2 xbLxQxg −= . B. Given g(x), and assume a smooth a(x), we still require two extra pieces of

information (initial/boundary data). We know ( )gLQxB −= 2/34)( , so we

need to solve )(41 2

2

2

xBdxda

adxad

=⎟⎠⎞

⎜⎝⎛+ , if 4/5)(: xah = , then solve

5/12

2

)(45 hxB

dxhd= .

Initial Considerations on Morphology Determination

1. Dendritic tree is passive ⇒ 2

2

)(xvvxg

tv

∂∂

=+∂∂

(uniform

properties). Our starting point, not the wave equation. 2. This leads to the spectral problem

λϕϕϕ=+− )(2

2

xgdxd

in rV\Ω

3. Classically, for all Ω∂∈γ , 0)( =γϕ .

4. Associated with operator gdxd

+− 2

2

is the discrete spectrum

{ } 1≥kkλ ; with associated eigenfunctions { } 1≥kkϕ , consider the pairs

1

|,≥

Γ⎭⎬⎫

⎩⎨⎧

k

kk dx

dϕλ = (Dirichlet) spectral data of graph Ω

Theorem (Belishev): If the spectral data of two trees coincide, the trees are spatially isometric. Remark: In determining the graph and densities g on Ω , Belishev (2004) studied the boundary controllability of the wave equation defined on rV\Ω and exploited that construction to recover the tree from its spectral data. Dynamic Inverse Problem ⎯⎯⎯ →⎯BCmethod Inverse Spectral Problem Remark (with Avdonin): For our case, need Neumann-to-Dirichlet map (inverse of Belishev’s), Neumann spectral data, comparable Spectral Inverse Problem, and develop BC theory from diffusion equation standpoint, rather than a wave equation standpoint. Part of the program has been done with the heat equation, recovering a diffusivity coefficient on a single interval domain (Avdonin, Belishev, Rozhkov, 1997).

Comments on “Swiss Cheese”* Project 1. Threshold behavior: special cases of conditional comparison principles for systems; prove stronger threshold results; propagation results for systems on tree graphs; explore relationship between upper and lower bounds on propagation speed ( cc , ) and model parameters 2. More general conditions with respect to graph, system dynamics for conduction block 3. Distal current sources: find a way to make progress for active networks (nonlinear cable theory), and tie to stronger propagation results 4. Finish channel recovery problem for branch and network problems; develop BC theory for cable theory on networks (new spectral inverse problem, etc.); develop numerical approach for highlighted inverse problems * “Swiss Cheese” because so many unfinished pieces of analysis and numerical work to do