analytics.wxm

/* [wxMaxima batch file version 1] [ DO NOT EDIT BY HAND! ]*/
/* [ Created with wxMaxima version 11.08.0 ] */

/* [wxMaxima: title   start ]
network representations 
of structured flows on manifolds.
   [wxMaxima: title   end   ] */

/* [wxMaxima: input   start ] */
imports: [implicit_plot, distrib, newton1]$
map(load, imports)$
ratprint: false $
forget_assumptions():=map(forget, assumptions)$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: section start ]
Phase representation of the FitzHugh Nagumo 
oscillator 
   [wxMaxima: section end   ] */

/* [wxMaxima: subsect start ]
reduction and analysis
   [wxMaxima: subsect end   ] */

/* [wxMaxima: comment start ]
describe fhn/excitator equations
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
dx : x(t)-x(t)^3/3+y(t);
dy : I-x(t);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
convert to radius, average phase, solve for steady state r
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
diff( sqrt(x(t)^2 + y(t)^2), t ) $ 
ev(%, [diff(x(t),t)= dx, diff(y(t), t)= dy]) $
ev(%, [x(t)=r*cos(p), y(t)=r*sin(p)]) , radcan, trigreduce $ 
integrate(%, p, 0, 2*%pi), factor;
solve([%=0], r);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
obtain phase equation, keeping lowest term
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
diff( atan( y(t)/x(t) ), t);
ev(%, [diff(x(t),t)= dx, diff(y(t), t)= dy]);
ev(%, [x(t)=r*cos(p), y(t)=r*sin(p), r=2]), radcan, trigsimp, factor, trigreduce;
dp:ev(%, [sin(4*p)=0, sin(2*p)=0]), factor;
wxplot2d(ev(%, [I=2]), [p, -%pi, %pi]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
fixed point existence and stability, first and third quandrants
are stable, where p and I have same sign. acos(2/I) gives correct
f.p. for I>0, not for I<0. 
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
dp_ss: rhs(solve([dp=0], p)[1]);
plot2d(dp_ss, [I, -4, 4]);
dp_l: diff(dp, p);
dp_l2: dp_l, p=dp_ss;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
obtain oscillation period by integrating dt/dphase from 
-pi to pi. 
answer that (I-2)(I+2) is positive (which corresponds to 
the oscillatory regime I<2
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
dp;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
answer negative here:
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
assume(4-I^2>0);
T0: integrate(1/dp, p, %pi, -%pi);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
simplify answer, change to rate
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
R1: 1/T0;
wxplot2d(if I^2<4 then R1 else 0, [I, -4, 4]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: subsect start ]
simulation
   [wxMaxima: subsect end   ] */

/* [wxMaxima: input   start ] */
dp;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
dp1: I*sin(p)/2 - 1;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
set_random_state( make_random_state(42) )$
dp_vf (p, I) := block(
  mode_declare(p, float, I, float),
  ev(-dp1))$
translate(dp_vf)$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
sol: []$ pi: -0.5$ dt: 0.05$ PI:ev(%pi, numer)$
jump_count: 0$

for i: 1 thru 100/dt do block(
  pi: pi + dt*(1 + 1.9*sin(pi)/2),
  if pi > %pi then block(pi: pi-2*PI, jump_count: jump_count + 1),
  if mod(i,5)=0 then sol: append(sol, [[dt*i, pi]])
)$
wxplot2d([discrete, sol]);
display(jump_count)$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: subsect start ]
prc
   [wxMaxima: subsect end   ] */

/* [wxMaxima: comment start ]
!!!!!!!!!!!!!!!!!!!!!!1 attention: Maxima will switch the integration limits when it wants to, so
that's why there are negative signs inserted by hand in order to make the results correct!!!!!!!!!!!
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
dp: (omega + g*delta)*cos(theta)/2 - 1;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
assume(omega^2<1);
dpa: 1 + sin(theta)*(omega + delta*g);
T0a: integrate(ev(1/dpa, delta=0), theta, -%pi, %pi);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
to both questions answer negative
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
assume(omega^2-4<0);
T0: integrate(ev(1/dp, [delta=0]), theta, %pi, -%pi);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
ts: T0*(-phi + %pi)/(2*%pi);
tsa: T0a*(phi + %pi)/(2*%pi);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
answer negative to this one, phi is always between -pi and pi
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
assume(phi<%pi) $
ts1: -integrate(ev(1/dp, [delta=0]), theta, phi, %pi), radcan;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
ts1: integrate(ev(1/dp, [delta=0]), theta, %pi, phi), radcan;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
ts1a: integrate(ev(1/dpa, [delta=0]), theta, -%pi, phi), radcan;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
need to ensure that in this plot, it starts at a reasonable positive number and goes down to zero
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
ts1, omega=1.;
wxplot2d([%, ev(ts1a, omega=0.5)], [phi, -%pi, %pi]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
g cos(phi) + 2*phi + 2*pi is positive when g < 0; starts to be neg from -pi and root moves toward pi as g increases away from 0.
g cos(phi) + 2*phi - 2*pi is negative when g > 0; starts to be pos from pi and root moves toward -pi as g decreases away from 0.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
trpp: -integrate(ev(1/dp, [delta=0]), theta, %pi, phi+g*cos(phi)/2), radcan;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
neg pos, pos neg and neg neg all give the same answer, so just use nn¦
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
trnn: -integrate(ev(1/dp, [delta=0]), theta, %pi, phi+g*cos(phi)/2), radcan;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
trnn: integrate(ev(1/dp, [delta=0]), theta, phi+g*cos(phi)/2, -%pi), radcan;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
trnna: integrate(ev(1/dpa, [delta=0]), theta, phi + g*sin(phi), %pi), radcan;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
trnn, omega=1.0 $
wxplot2d([ev(%, g=0.5), ev(%, g=-0.5)], [phi, -%pi, %pi]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
trnna, omega=0.5 $
wxplot2d([ev(%, g=0.5), ev(%, g=-0.5)], [phi, -%pi, %pi]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
T1nn: ts1 + trnn, radcan;
T1pp: ts1 + trpp, radcan;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
[T0, T1pp, T1nn], g=-0.0, omega=1., numer, phi=0.0;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
(ts1 + trnn)/T0 - 1, radcan $
prc: %;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
omega:'omega;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
ts1a + trnna, radcan;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
T0a;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
(ts1a + trnna)/T0a - 1, radcan $
prca: %;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
wxplot2d(ev(prc, omega=1.0, g=-0.5), [phi, -%pi, %pi]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
wxplot2d(ev([prca, diff(prca, phi)], omega=0.5, g=0.5), [phi, -%pi, %pi]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
bd : ev(1.0*%pi, numer);
C1: ev(ts1a, omega=1/2-dw, phi=phi1) = ev(trnna, omega=1/2+dw, phi=phi2) $
C2: ev(ts1a, omega=1/2+dw, phi=phi2) = ev(trnna, omega=1/2-dw, phi=phi1) $
lambda : (diff(ev(prca,phi=phi1,omega=1/2-dw), phi1) - 1)*(diff(ev(prca,phi=phi2,omega=1/2+dw), phi2) - 1) $
pars : [g=0.5, dw=0.] $
implicit_plot(ev([C1, C2], pars), [phi1, -bd, bd], [phi2, -bd, bd]);
plot3d(ev(lambda, pars), [phi1, -bd, bd], [phi2, -bd, bd]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
T0a;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
(rhs(C1)-lhs(C1))/T0a, radcan;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
wxplot2d(ev([(rhs(C1)-lhs(C1))/T0a, (rhs(C2)-lhs(C2))/T0a, lambda-1], 
    omega=1/2, phi1=phi, phi2=phi, g=0.3, dw=0.), [phi, -3.16, 3.16]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
omega: 0.6 $ dw: 0.000$
o1: omega+dw $
o2: omega-dw $
Tt: ev(T0a, omega=o1)/ev(T0a, omega=o2), numer $
R0s: [1000/ev(T0a, omega=o1), 1000/ev(T0a, omega=o2)], numer;
Ttinv: 1/Tt $
f1: ev(prca, omega=o1) $
f2: ev(prca, omega=o2) $
dphi: Ttinv*(1 + ev(f2, phi=Tt*(1 - ts1a/T0a + f1))) - (1 + f1) $
wxplot2d([(R0s[1]/10)*ev(dphi, g=0.1), ev(dphi, g=0.1)], [phi, -%pi, %pi], ['y, -%pi, %pi]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
dprca: diff(prca, phi) $
wxplot2d([ev(dprca, omega=1/2, g=0.2)-1, 
          ev(dprca, omega=1/2, g=-0.2)-1], [phi, -%pi, %pi],
    [legend, "g>0", "g<0"]);
wxplot2d(ev(dprca, omega=1/2, phi=%pi-0.001)-1, [g, -1, 1]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
Z(phis, N) := abs(sum(exp(%i*phis[k]), k, 1, N)/N) $
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
n: 2;
prc, g=n*g*Z(phi, n), phi=sum(phi[k],k,1,n)/n $
diff(%, phi[2]) $
ev(%, makelist(phi[i]=%pi, i, 1, n));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
single eigenvalue for mode formation, evaluted at synchronization fixed point
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
Z(phis, N) := abs(sum(exp(%i*phis[k]), k, 1, N)/N) $
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
n: 6;
prc, g=n*g*Z(phi, n), phi=sum(phi[k],k,1,n)/n $ diff(%, phi[2]) $
fp: ev(%, makelist(phi[i]=%pi, i, 1, n));
plot3d(min(abs(fp),1) , [g, -%pi/2, %pi/2], [omega, 0, 2]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
exp(%i*pi), numer;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
2nd order approx around 0 point is good enough for local work
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
ratprint:false;
gmaster: %pi/2;
fix: ev(prc, [omega=1+mo, g=gmaster, phi=%pi+mp]) - ev(prc, [omega=1-mo, g=gmaster, phi=%pi-mp]) $
t1: taylor(fix, [mo, mp], 0, 1) $
t3: taylor(fix, [mo, mp], 0, 3) ;
f: diff(prc, phi) $ 
f1: ev(f, [omega=1 + mo, phi=%pi+mp, g=gmaster]) $ 
f2: ev(f, [omega=1 - mo, phi=%pi-mp, g=gmaster]) $ 
lambda : (f1-1)*(f2-1) $
lam1: taylor(lambda, [mo, mp], 0, 2), radcan $ ev(%, numer);
solve([diff(lam1, mp)=0, diff(lam1, mo)=0], [mp, mo]) ; ev(lam1, %, numer);

wximplicit_plot([fix=0, t3=0, abs(lam1)=1.0, abs(lambda)=1.0], 
[mo, -0.5, 0.5], [mp, -1.0, 1.0], [legend,""]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
t3 $ ev(%, numer, expand)/0.0852811381638135 $ ev(%, expand, numer);
lam1 $ ev(%, numer, expand)/-0.27217101584925 $ ev(%, expand, numer);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
solve([t1=0], mp), expand;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
now if I build a matrix for synchronization of n neurons:
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
N: 4;
F: apply(matrix, makelist(makelist(f[j], j, 1, N), i, 1, N))$
Is: apply(matrix, makelist(makelist(if (j=mod(i,N)+1) then 1 else 0, j, 1, N), i, 1, N)) $
F-Is;
eigenvalues(F-Is) $
makelist(float(ev(abs(float(%[1][i])),numer)), i, 1, N) $
%[1];
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
prca, omega=1/2, 
/* [wxMaxima: input   end   ] */

/* [wxMaxima: section start ]
Two theta neurons coupled
   [wxMaxima: section end   ] */

/* [wxMaxima: subsect start ]
rate dynamics
   [wxMaxima: subsect end   ] */

/* [wxMaxima: comment start ]
We know the rate of a theta neuron, but inorder to endow 'rate' with 
indendent dynamics, we want a simple dynamics that follows this.
   [wxMaxima: comment end   ] */

/* [wxMaxima: comment start ]
first, try to average over input to match the steady state of a 
nonlinear synapse 
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
R2: (I^2-4)^alpha/(4*%pi);
assume(alpha>0) $
R2_Iavg: integrate( ev(R2, [(I^2-4) = abs(I^2-4)]), I, 0, 2);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
R3: rhs(solve([0 = (b - r3)*H - r3/a], r3)[1]);
assume(a>0) $
R3_Havg: integrate(R3, H, 0, 4^alpha);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
this doesn't go very far, so let's assume synaptic dynamics
are linear, follow adiabatically
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
ev(solve([0 = H - 'r], 'r), [H=R2]) $
ev(%, [alpha=1]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
so now, the in coupled case (need to look at notes again)
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
ev(R2 - r[i], [alpha=1, I=(sum(g[i,j]*r[j], j, 1, N))]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
As my notes confirm, I assumed the use of linear dynamics. Bon: 
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
R1;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
rdot: ;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
If the two neurons or modes are mutually and symmetrically but not self coupled, 
there is a symmetry w.r.t. to change of variables in their steady states
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
assumptions : [r2>0, r2^2-4>0] $
map(assume, assumptions) $

ev('diff(r, t)*4*%pi = sqrt(abs(I^2-4)) - 'r, [r=r1, I=r2]);
%, 'diff(r1, t)=0 $ solve(%, r1);
%, r1=r2, r2=r1;
solve(%[1], r1);

map(forget, assumptions) $
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Here we address the possible manifold in a two dimensional system, but this is not
a particularly general result? 

- Now address the flows, and generalize to N-neuron modes? 
- Also have to address excitator dynamics in two neuron/mode networks. 
   [wxMaxima: comment end   ] */

/* [wxMaxima: subsect start ]
Phase dynamics
   [wxMaxima: subsect end   ] */

/* [wxMaxima: comment start ]
What we can consider here is the situation of synchronization of two pulse coupled oscillators 
at two different (but temporally stable) frequencies. We should see (and show) more or less
HKB style dynamics, Arnold tongues. Here's the article to look at: 

deGuzman and Kelso - Multifrequency behavioral patterns and the phase attractive circle map
   [wxMaxima: comment end   ] */

/* [wxMaxima: section start ]
Three neurons
   [wxMaxima: section end   ] */

/* [wxMaxima: subsect start ]
firing rate
   [wxMaxima: subsect end   ] */

/* [wxMaxima: comment start ]
1. project to xi coords
2. SIG-S exchange
3. to phi, r, r coords

The important part is to arrive again at three results: 

1. existence & stability of normal (1, 1, 1) dynamics
2. existence & stability of radial dynamics
3. existence & stability of heteroclinic cycle phase dynamics. 
   [wxMaxima: comment end   ] */

/* [wxMaxima: comment start ]
analysis of connectivity matrix creating 3-cycle, assuming we get throw the sqrt

We'd like to project this onto the normal, radial and phase coordinates, so 
we make a guess using the right singular vectors of the connectivity with g=0 mu=1; 
with the reverse projection; define derivatives

obtain rate and normal dynamics, phase averaged; evaluate phase dynamics with solution
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
asymm: matrix([ 0, -1,  1], 
              [ 1,  0, -1], 
              [-1,  1,  0] ) $
G: g-g*ident(3)+mu*asymm;

apply(matrix, makelist([r[i]], i, 1, 3))$ Grs: G . % $
drs: apply(matrix, makelist([(Grs[i][1]^2 - 4) - r[i]], i, 1, 3));

[U, S, VT] : dgesvd(ev(G, [g=0, mu=1]), false, true) $
minus1(x) := if x>0 then 1 else -1 $
VT: matrixmap(
  lambda([mij], if abs(mij)>1e-2 then minus1(mij)*sqrt(2/round(2/(mij^2))) else 0),
  VT ), radcan ;

matrix([x(t)], [y(t)], [n(t)]) $ transpose(VT) . % $
rx_reverse_proj: makelist(r[i]=%[i][1],i,1,3) $

dx: ev((VT . drs)[1][1], rx_reverse_proj)$
dy: ev((VT . drs)[2][1], rx_reverse_proj)$
dn: ev((VT . drs)[3][1], rx_reverse_proj)$

xyn2prn: matrix([-(sin(p)*r)/(sin(p)^2*r^2+cos(p)^2*r^2),
               (cos(p)*r)/(sin(p)^2*r^2+cos(p)^2*r^2),0],
                 [(2*cos(p)*r)/(2*sqrt(sin(p)^2*r^2+cos(p)^2*r^2)),
                  (2*sin(p)*r)/(2*sqrt(sin(p)^2*r^2+cos(p)^2*r^2)), 0],
                 [0, 0, 1])$
VT . drs, rx_reverse_proj, radcan, n(t)=n $ %, x(t)=r*cos(p), y(t)=r*sin(p) $
DPRN: xyn2prn . %, expand, trigreduce, radcan;

r_avg: integrate(DPRN[2][1], p, -%pi, %pi) $
n_avg: integrate(DPRN[3][1], p, -%pi, %pi) $
nr_avg: solve([r_avg=0, n_avg=0], [n, r])[2], factor;

NPI: %pi, numer $
DPRN[1][1], nr_avg $ pl: diff(%, p), g=-1/2;
ev(DPRN[1][1], nr_avg), factor $ pd: %, g=-1/2,  radcan;
wximplicit_plot([%=0, pl=0, pl=-1/3], [p, -NPI, NPI], [mu, -0.3, 0.3], [legend,""]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
pd;
pl, factor;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
asymm: matrix([ 0, -1,  1], 
             [ 1,  0, -1], 
              [-1,  1,  0] ) $
G: g-g*ident(3)+mu*asymm$

G[1,2]: g - mu*(1-alpha) $ 
G[2,1]: g + mu*(1-alpha) $ display(G)$

apply(matrix, makelist([r[i]], i, 1, 3))$ Grs: G . % $
drs: apply(matrix, makelist([(Grs[i][1]^2 - 4) - r[i]], i, 1, 3))$

matrix([x(t)], [y(t)], [n(t)]) $ transpose(VT) . % $
rx_reverse_proj: makelist(r[i]=%[i][1],i,1,3) $

dx: ev((VT . drs)[1][1], rx_reverse_proj)$
dy: ev((VT . drs)[2][1], rx_reverse_proj)$
dn: ev((VT . drs)[3][1], rx_reverse_proj)$

xyn2prn: matrix([-(sin(p)*r)/(sin(p)^2*r^2+cos(p)^2*r^2),
                 (cos(p)*r)/(sin(p)^2*r^2+cos(p)^2*r^2),0],
                 [(2*cos(p)*r)/(2*sqrt(sin(p)^2*r^2+cos(p)^2*r^2)),
                  (2*sin(p)*r)/(2*sqrt(sin(p)^2*r^2+cos(p)^2*r^2)), 0],
                 [0, 0, 1])$
VT . drs, rx_reverse_proj, radcan, n(t)=n $ %, x(t)=r*cos(p), y(t)=r*sin(p) $
DPRN: xyn2prn . %,  radcan$

DPRN[2][1], radcan, sin(p)^2+cos(p)^2=1, expand, factor $
%, sin(p)^2+cos(p)^2=1, trigsimp, trigreduce$
r_avg: integrate(%, p, -%pi, %pi) $
n_avg: integrate(DPRN[3][1], p, -%pi, %pi)  $
nr_avg: solve([r_avg=0, n_avg=0], [n, r])[2], factor $

NPI: %pi, numer $
DPRN[1][1], nr_avg $ pl: diff(%, p), g=-1/2 $
dp : ev(DPRN[1][1], nr_avg), factor $ %, g=-1/2, radcan $
dp1: dp, g=-1/2, factor, sin(p)^2 + cos(p)^2=1, trigsimp, radcan $
dp2: dp1, factor, trigsimp, trigreduce, trigrat, ratsimp, radcan ;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
tex(dp2);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
dp2;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
match( a(b*sin + c*cos)+d, dp2) $
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
isolating teh terms of the sin and cos components
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
dp2, expand $ isolate(%, sin(3*p)) $ %, radcan $
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
isolate(dp2, sin(3*p)) $
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
isolate(%t986, sin(2*p));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
isolate(%t994,  cos(2*p));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
isolate(%t984, cos(3*p));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
isolate(%t989, sin(p));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
isolate(%t991, cos(p));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
dp1: dp, mu=1, g=-1/2, alpha=2/3, radcan $
dp2: dp, mu=1, g=-1/2, alpha=0, radcan $
dp3: dp, mu=0, g=-1/2, alpha=0, radcan $
wxplot2d([dp1, dp2, dp3], [p, -%pi, %pi],
 [legend, "mu=1, alpha=1/3", 
"mu=1", 
"mu=0"])$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
warning: this integration takes 5 minutes
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
dp, g=-1/2, radcan, expand, trigreduce $
integrate(%, p, -%pi, %pi);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
plot3d(%, [mu, -2, 2], [alpha, -5, 3]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
here we can say we've captured it: existence and stability of phase fixed points as asymmetry
varies. For small asymmetry the fixed points remain, while larger asymmetry produces cycling.
We also show the effect of greater symmetry breaking: going between multistability and limit
cycle to obtain monostability. 
   [wxMaxima: comment end   ] */

/* [wxMaxima: section start ]
Four and more : ensemble couplings
   [wxMaxima: section end   ] */

/* [wxMaxima: comment start ]
We maybe should not dwell too much here except to demosntrate some stability 
properties of ensembles. 
   [wxMaxima: comment end   ] */

/* [wxMaxima: subsect start ]
mode formation
   [wxMaxima: subsect end   ] */

/* [wxMaxima: input   start ] */
dq(i,n):= ev(-q[i] + sqrt(1 - (g*sum(q[j], j, 1, n) + I)^2), sumexpand);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
n: 3;
makelist( makelist(diff(dq(i, n), q[j]), j, 1, n), i, 1, n) $
apply(matrix, %) $
evs: eigenvalues(%), radcan;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
n: m $
w0p: (-g*n*I + sqrt(g^2*n^2 - I^2 + 1))/(g^2*n^2 + 1) $
lam: 1 - n*g*(I + g*n*w)/sqrt(1 - g*n*w) $
lam, w=w0p $ %, I=0 ;
wximplicit_plot([%=0], [g, -1, 1], [m, 2, 30], [legend, ""]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
evs[1][2];
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
evs[1][2];
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
(-I+(-q[3]-q[2]-q[1])*g+2)*(I+(q[3]+q[2]+q[1])*g+2), factor;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
4-(g*sum(q[i],i,1,3) + I)^2, expand, factor;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: subsect start ]
Two and two
   [wxMaxima: subsect end   ] */

/* [wxMaxima: subsect start ]
Two and three
   [wxMaxima: subsect end   ] */

/* [wxMaxima: subsect start ]
Three and three
   [wxMaxima: subsect end   ] */

/* [wxMaxima: section start ]
Excitator dynamics
   [wxMaxima: section end   ] */

/* [wxMaxima: subsect start ]
Original formulation
   [wxMaxima: subsect end   ] */

/* [wxMaxima: subsect start ]
Simulating the Excitator
   [wxMaxima: subsect end   ] */

/* [wxMaxima: input   start ] */
rng_state: make_random_state(42)$
set_random_state(rng_state);
load(distrib);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
sig: 2$
foo(t):=random_normal(0., sig+t/1000000.0)$
dx : x - x^3/3 + y + foo(t)$
dy : 0.3 -x + foo(t)$
sol: rk([dx, dy], [x, y], [0, 1], [t, 0, 30, 0.05])$
wxplot2d([discrete, map(lambda([x],[x[1],x[2]]), sol)]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: subsect start ]
Generalizing the Excitator constraints
   [wxMaxima: subsect end   ] */

/* Maxima can't load/batch files which end with a comment! */
"Created with wxMaxima"$