Increased computational accuracy in multi-compartmental cable models (Lindsay et al. 2005)

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Accession:129149
Compartmental models of dendrites are the most widely used tool for investigating their electrical behaviour. Traditional models assign a single potential to a compartment. This potential is associated with the membrane potential at the centre of the segment represented by the compartment. All input to that segment, independent of its location on the segment, is assumed to act at the centre of the segment with the potential of the compartment. By contrast, the compartmental model introduced in this article assigns a potential to each end of a segment, and takes into account the location of input to a segment on the model solution by partitioning the effect of this input between the axial currents at the proximal and distal boundaries of segments. For a given neuron, the new and traditional approaches to compartmental modelling use the same number of locations at which the membrane potential is to be determined, and lead to ordinary differential equations that are structurally identical. However, the solution achieved by the new approach gives an order of magnitude better accuracy and precision than that achieved by the latter in the presence of point process input.
Reference:
1 . Lindsay AE, Lindsay KA, Rosenberg JR (2005) Increased computational accuracy in multi-compartmental cable models by a novel approach for precise point process localization. J Comput Neurosci 19:21-38 [PubMed]
Model Information (Click on a link to find other models with that property)
Model Type: Neuron or other electrically excitable cell;
Brain Region(s)/Organism:
Cell Type(s):
Channel(s): I Na,t; I K;
Gap Junctions:
Receptor(s):
Gene(s):
Transmitter(s):
Simulation Environment: NEURON; C or C++ program;
Model Concept(s): Methods;
Implementer(s):
Search NeuronDB for information about:  I Na,t; I K;
/
LindsayEtAl2005
readme.txt
03-192.pdf
AnalyseResults.c
BitsAndPieces.c
CellData.dat
CompareSpikeTrain.c
Ed04.tex
ExactSolution.dat
GammaCode
Gen.tex
Gen1.tex
Gen2.tex
Gen3.tex
Gen4.tex
Gen5.tex
Gen6.tex
GenCom.c
GenCom1.c
GenCom2.c
GenComExactSoln.c
GenerateInput.c
GenerateInputText.c
GenRan.ran
GetNodeNumbers.c
Info100.dat
Info20.dat
Info200.dat
Info30.dat
Info300.dat
Info40.dat
Info400.dat
Info50.dat
Info500.dat
Info60.dat
Info70.dat
Info80.dat
Info90.dat
InputCurrents.dat
InputDendrite.dat
JaySpikeTrain.c
JayTest1.dat
JayTest100.dat
KenSpikeTrain.c
KenTest1.dat *
KenTest10.dat
KenTest100.dat *
KenTest10p.dat
KenTest1p.dat *
KenTest2.dat
KenTest2p.dat
KenTest3.dat
KenTest3p.dat
KenTest4.dat
KenTest4p.dat
KenTest5.dat
KenTest5p.dat
KenTest6.dat
KenTest6p.dat
KenTest7.dat
KenTest7p.dat
KenTest8.dat
KenTest8p.dat
KenTest9.dat
KenTest9p.dat
LU.c
Mean50.dat
Mean500.dat
mosinit.hoc
NC.pdf
NC.tex
NC1.tex
NC2.tex
NC3.tex
NC4.tex
NC5.tex
NC6.tex
NCFig2.eps *
NCFig3.eps *
NCFig4.eps *
NCFig5a.eps *
NCFig5b.eps *
NCFig6.eps *
NCPics.tex
NeuronDriver.hoc
NewComExactSoln.c
NewComp.pdf
NewComp.ps
NewComp.tex
NewComp.toc
NewComp1.tex
NewComp2.tex
NewComp3.tex
NewComp4.tex
NewComp5.tex
NewComp6.tex
NewCompFig1.eps
NewCompFig2.eps *
NewCompFig3.eps *
NewCompFig4.eps *
NewCompFig5a.eps *
NewCompFig5b.eps *
NewCompFig6.eps *
NewCompPics.tex
NewComSpikeTrain.c
NewRes.dat
NewRes60.dat
NewRes70.dat
NewRes80.dat
NewSynRes40.dat
NewTestCell.d3
NResults.res
OldComExactSoln.c
out.res
principles_01.tex
rand
Ratio.dat
RelErr.dat
ReviewOfSpines.pdf
SpikeTimes.dat
TestCell.d3
TestCell1.d3
TestCell2.d3
TestCell3.d3
TestCell4.d3
testcellnew2.hoc
TestCGS.c
TestGen1.c
TestSim.hoc
TestSim020.hoc
TestSim030.hoc
TestSim040.hoc
TestSim050.hoc
TestSim060.hoc
TestSim070.hoc
TestSim080.hoc
TestSim090.hoc
TestSim1.hoc
TestSim100.hoc
TestSim200.hoc
TestSim300.hoc
TestSim400.hoc
TestSim500
TestSim500.hoc
                            
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <string.h>


 /***************************************************************
    Traditional compartmental model with large scale synaptic
    input and active soma. Spike generating program
  ***************************************************************/

/* Global definitions */
#define             CS    1.0
#define             GS    0.091
#define             GA    14.286
#define             CM    1.0
#define             GM    0.091
#define         OUTPUT    "JaySpike300.dat"
#define           TEND    11             /* Must be an integer */
#define           GMAX    3.0e-5
#define           RATE    30.0
#define           VSYN    115.0
#define             NT    1000
#define          NODES    300
#define          NSEED    2             /* Seed for random number generator */
#define        CELSIUS    18.5          /* Celsius temperature of neuron */
#define            CGS    1.0e-24       /* Tolerance used in CGS algorithm */
#define            TAU    0.5

/* Parameters for exact solution */
#define           NSYN    100          /* Number of synapses */
#define             RS    0.005


typedef struct SparseMatrix_t
{
        double *a;
        int *col;
        int *StartRow;
        int n;
} SparseMatrix;


typedef struct cond_t
{
        int n;                      /* Number of time steps in conductance profile */
        double dt;                  /* Integration time step (msec) */
        double *g;                  /* Conductance profile for prescribed dt (mu a) */
        double tol;                 /* Determines off threshold for synapse */
        double tau;                 /* Rise time for synapse (msec) */
        double gmax;                /* Peak conductance per synapse */
} cond;


typedef struct synapse_t
{
        int id;                     /* Identifies contact type */
        double xp;                  /* Location of contact */
        int cn;                     /* Central node of compartment*/
        int diag;                   /* Diagonal entry in sparse matrix corresponding to cn */
        double gval;                /* Present conductance */

/* Properties of synaptic conductance profile */
        cond *SynCond;              /* Address of synaptic conductance profile */
        int SynTime;                /* Time to activation of syapse/time since activation */
        int MaxStep;                /* Time to inactivation of synapse */
        double vsyn;                /* Reversal potential of synaptic species */
        struct synapse_t *NextSyn;  /* Address of next synapse */
} synapse;


typedef struct branch_t
{
/*  Connectivity of branch */
        struct branch_t *parent;    /* Pointer to parent branch */
        struct branch_t *child;     /* Pointer to child branch */
        struct branch_t *peer;      /* Pointer to a peer branch */

/*  Physical properties of branch */
        int id;                     /* Branch identifier */
        double xl;                  /* X-coordinate of lefthand endpoint */
        double yl;                  /* Y-coordinate of lefthand endpoint */
        double zl;                  /* Z-coordinate of lefthand endpoint */
        double xr;                  /* X-coordinate of righthand endpoint */
        double yr;                  /* Y-coordinate of righthand endpoint */
        double zr;                  /* Z-coordinate of righthand endpoint */
        double diam;                /* Branch diameter (cm) */
        double plen;                /* Branch length (cm) */
        double hseg;                /* Compartment length (cm) */

/*  Node information for spatial representation */
        int nc;                     /* Number of compartments in branch */
        int junct;                  /* Junction node of the branch */
        int first;                  /* Internal node connected to junction */

/*  Synapse information */
        synapse *synlist;           /* Branch contact */
} branch;


typedef struct dendrite_t
{
        branch *root;               /* Pointer to root branch of dendrite */
        double plen;               /* Total length of dendrite */
} dendrite;


typedef struct neuron_t
{
        int ndend;                  /* Number of dendrites */
        dendrite *dendlist;         /* Pointer to an array of dendrites */
} neuron;


/* Function type declarations */
int     Count_Branches( branch *, branch *),
        Count_Synapses( branch *, branch *);

cond *ConductanceProfile( double, double,  double, double );

double  branch_length( branch *, branch *),
        ran(unsigned int *, unsigned int *, unsigned int *),
        alfa_h( double ),
        alfa_m( double ),
        alfa_n( double ),
        beta_h( double ),
        beta_m( double ),
        beta_n( double );

void    Remove_Branch( branch **, branch *),
        Output_Information( branch *, FILE *),
        Assign_Branch_Nodes( branch *, double *),
        Enumerate_Nodes( branch *, int *),
        Matrix_Vector_Multiply( SparseMatrix *, double *, double *),
        Matrix_Malloc( SparseMatrix *, int, int),
        Matrix_Free( SparseMatrix *),
        Generate_Dendrite(branch *, int *),
        Build_Test_Dendrite( branch **, branch *),
        Initialise_Synapses( branch *),
        Update_Synapses( branch *),
        cgs( int *, SparseMatrix *, double *, double *, double);

/* Global Variables */
SparseMatrix lhs, rhs;
double pi, dt, *NewAmp;
unsigned int ix, iy, iz;

int main( int argc, char **argv )
{
    extern unsigned int ix, iy, iz;
    extern double pi, dt, *NewAmp;
    extern SparseMatrix lhs, rhs;
    int j, k, nodes, n, nn, FirstNode, repeat, maxstep, spk, nspk,
        nstep, nb, nsyn, getmem, connected, counter, NumberOfSynapses;
    double *v, *x, *OldAmp, AreaOfSoma, tmp, h, len, CellLength, tnow,
        gs, aval, bval, hval, nval, mval, vs, sc, vm0, vm1, vm2,
        LocusContact;
    void srand( unsigned int);
    double *StoredLHS, *StoredRHS;
    double v_na=115.0, v_k=-12.0, v_l, g_na=120.0, g_k=36.0, g_l=0.3;
    synapse *newsyn, *syn;
    neuron *cell;
    cond *SynCond;
    branch *bold, *bnew, *FirstBranch;
    char word[100];
    FILE *fp, *fp1;

/*  Initialise random number generator */
    fp = fopen(OUTPUT,"w");
    fclose(fp);
    nspk = spk = 0;
    dt = 1.0/((double) NT);
    srand( ((unsigned int) NSEED) );
    ix = rand( );
    iy = rand( );
    iz = rand( );
    SynCond = ConductanceProfile( dt, TAU, 5.e-7, GMAX );

/*  Load Test Neuron */
    maxstep =  1000*NT*TEND;
    pi = 4.0*atan(1.0);
    AreaOfSoma = 4.0*pi*RS*RS;
    if ( argc != 2 ) {
        printf("\n Invoke program with load <input>\n");
        return 1;
    }
    printf("\nOpening file %s\n",argv[1]);
    if ( (fp=fopen(argv[1],"r")) == NULL ) {
        printf("\n Test Neuron file does not found");
        return 1;
    }

/*  Get branch data */
    bold = NULL;
    while  ( fscanf(fp,"%s", word) != EOF ) {
        if ( strcmp(word, "Branch") == 0 || strcmp(word, "branch") == 0 ) {
            fscanf(fp, "%d", &nodes);
            printf("Found branch %d \n", nodes);
            bnew = (branch *) malloc( sizeof(branch) );
            bnew->id = nodes;
            bnew->peer = NULL;
            bnew->child = NULL;
            bnew->synlist = NULL;
            if ( bold != NULL) {
                bold->child = bnew;
            } else {
                FirstBranch = bnew;
            }
            bnew->parent = bold;
            fscanf(fp,"%lf %lf %lf", &(bnew->xl), &(bnew->yl), &(bnew->zl));
            fscanf(fp,"%lf %lf %lf", &(bnew->xr), &(bnew->yr), &(bnew->zr));
            fscanf(fp,"%lf %lf", &(bnew->plen), &(bnew->diam));
            bold = bnew;
        } else {
            printf("Unrecognised dendritic feature\n");
            return 0;
        }
    }
    fclose(fp);

/*  Compute total length of dendrite */
    CellLength = 0.0;
    bnew = FirstBranch;
    while ( bnew ) {
       CellLength += bnew->plen;
       bnew = bnew->child;
    }

/*  STEP 1. - Randomly place NSYN synapses on branches */
    for ( k=0 ; k<NSYN ; k++ ) {
        LocusContact = CellLength*ran( &ix, &iy, &iz);
        bnew = FirstBranch;
        len = bnew->plen;
        while ( LocusContact > len ) {
            bnew = bnew->child;
            len += bnew->plen;
        }
        newsyn = (synapse *) malloc( sizeof(synapse) );
        newsyn->NextSyn = NULL;
        newsyn->SynCond = SynCond;
        newsyn->MaxStep = SynCond->n;
        newsyn->vsyn = VSYN;
        newsyn->xp = LocusContact-(len-bnew->plen);
        syn = bnew->synlist;
        if ( syn ) {
            if ( newsyn->xp < syn->xp ) {
                newsyn->NextSyn = syn;
                bnew->synlist = newsyn;
            } else {
                while ( syn->NextSyn && newsyn->xp > syn->NextSyn->xp ) syn = syn->NextSyn;
                if ( syn->NextSyn ) newsyn->NextSyn = syn->NextSyn;
                syn->NextSyn = newsyn;
            }
        } else {
            bnew->synlist = newsyn;
        }
    }

/*  STEP 2. - Count root branches */
    bold = FirstBranch;
    n = 0;
    while ( bold ) {
        bnew = FirstBranch;
        do {
            tmp = pow(bold->xl-bnew->xr,2)+
                  pow(bold->yl-bnew->yr,2)+
                  pow(bold->zl-bnew->zr,2);
            connected = ( tmp < 0.01 );
            bnew = bnew->child;
        } while ( bnew && !connected );
        if ( !connected ) n++;
        bold = bold->child;
    }

/*  STEP 3. - Identify somal dendrites but extract nothing */
    printf("\n\nTree contains %d individual dendrite(s) ...\n", n);
    cell = (neuron *) malloc( sizeof(neuron) );
    cell->ndend = n;
    cell->dendlist = (dendrite *) malloc( n*sizeof(dendrite) );
    bold = FirstBranch;
    n = 0;
    while ( n < cell->ndend ) {
        bnew = FirstBranch;
        do {
            tmp = pow(bold->xl-bnew->xr,2)+
                  pow(bold->yl-bnew->yr,2)+
                  pow(bold->zl-bnew->zr,2);
            connected = ( tmp < 0.01 );
            bnew = bnew->child;
        } while ( bnew && !connected );
        if ( !connected ) {
            cell->dendlist[n].root = bold;
            n++;
        }
        bold = bold->child;
    }

/*  STEP 4. - Extract root of each dendrite from dendrite list */
    for ( k=0 ; k<cell->ndend ; k++ ) {
        bold = cell->dendlist[k].root;
        Remove_Branch( &FirstBranch, bold);
    }

/*  STEP 5. - Build each test dendrite from its root branch */
    for ( k=0 ; k<cell->ndend ; k++ ) {
        Build_Test_Dendrite( &FirstBranch, cell->dendlist[k].root );
    }
    if ( FirstBranch != NULL ) printf("\nWarning: Unconnected branch segments still exist\n");

/*  STEP 6. - Count number of synapses on Cell */
    NumberOfSynapses = 0;
    for ( k=0 ; k<cell->ndend ; k++ ) {
        bnew = cell->dendlist[k].root;
        NumberOfSynapses += Count_Synapses( cell->dendlist[k].root, bnew);
    }
    printf("\nNumber of Synapses %d", NumberOfSynapses);

/*  STEP 7. - Count dendritic segments */
    for ( nb=k=0 ; k<cell->ndend ; k++ ) {
        bnew = cell->dendlist[k].root;
        nb += Count_Branches( bnew, bnew);
    }
    h = CellLength/((double) NODES-nb);
    for ( k=0 ; k<cell->ndend ; k++ ) Assign_Branch_Nodes( cell->dendlist[k].root, &h);

/*  STEP 8. - Enumerate Nodes */
    FirstNode = 0;
    for ( k=0 ; k<cell->ndend ; k++ ) Enumerate_Nodes( cell->dendlist[k].root, &FirstNode );
    for ( k=0 ; k<cell->ndend ; k++ ) cell->dendlist[k].root->junct = FirstNode;
    printf("\nNumber of nodes is %d\n", FirstNode+1);

/*  STEP 9. - Construct Sparse Matrices */
    nodes = FirstNode+1;
    Matrix_Malloc( &lhs, nodes, 3*nodes-2 );
    Matrix_Malloc( &rhs, nodes, 3*nodes-2 );
    lhs.StartRow[0] = rhs.StartRow[0] = 0;
    for ( counter=k=0 ; k<cell->ndend ; k++ ) {
        bnew = cell->dendlist[k].root;
        Generate_Dendrite( bnew, &counter);
    }
    lhs.n = rhs.n = nodes;

/*  STEP 10. - Do somal node */
    lhs.a[3*nodes-3] = rhs.a[3*nodes-3] = 0.0;
    for ( k=0 ; k<cell->ndend ; k++ ) {
        bnew = cell->dendlist[k].root;
        lhs.a[counter] = 0.0;
        rhs.a[counter] = -0.25*pi*pow(bnew->diam,2)/(bnew->hseg);
        rhs.a[3*nodes-3] += 0.25*pi*pow(bnew->diam,2)/(bnew->hseg);
        lhs.a[3*nodes-3] += 0.5*pi*(bnew->diam)*(bnew->hseg);
        lhs.col[counter] = rhs.col[counter] = bnew->first;
        counter++;
    }
    lhs.col[3*nodes-3] = rhs.col[3*nodes-3] = nodes-1;
    lhs.StartRow[nodes] = rhs.StartRow[nodes] = 3*nodes-2;

/*  STEP 11. - Build and store constant components of left and right hand matrices  */
        for ( k=0 ; k<3*nodes-2 ; k++ ) {
        rhs.a[k] = 0.5*dt*(GA*rhs.a[k]+GM*lhs.a[k]);
        rhs.a[k] = CM*lhs.a[k]-rhs.a[k];
        lhs.a[k] = 2.0*CM*lhs.a[k]-rhs.a[k];
    }

/*  Add capacitive term of soma */
    lhs.a[3*nodes-3] += AreaOfSoma*CS;
    rhs.a[3*nodes-3] += AreaOfSoma*CS;
    StoredLHS = (double *) malloc( (3*nodes-2)*sizeof(double) );
    StoredRHS = (double *) malloc( (3*nodes-2)*sizeof(double) );
    for ( k=0 ; k<3*nodes-2 ; k++ ) {
        StoredLHS[k] = lhs.a[k];
        StoredRHS[k] = rhs.a[k];
    }

//    fp1 = fopen("JayOutput.dat","w");
//    for( k=0 ; k<3*nodes-2 ; k++ ) {
//        fprintf(fp1,"%12.10lf\n", lhs.a[k]);
//    }
//    fclose(fp1);
//    printf("\nOutput information");
//    getchar( );

/*  Step 12. - Initialise synapses, soma and membrane potentials */
    for ( k=0 ; k<cell->ndend ; k++ ) {
        bnew = cell->dendlist[k].root;
        Initialise_Synapses(bnew);
    }
    v = (double *) malloc( (nodes)*sizeof(double) );
    x = (double *) malloc( (nodes)*sizeof(double) );
    OldAmp = (double *) malloc( (nodes)*sizeof(double) );
    NewAmp = (double *) malloc( (nodes)*sizeof(double) );
    for ( k=0 ; k<nodes ; k++ ) v[k] = 0.0;

    g_na *= AreaOfSoma;
    g_k *= AreaOfSoma;
    g_l *= AreaOfSoma;
    hval = alfa_h(0.0)/(alfa_h(0.0)+beta_h(0.0));
    mval = alfa_m(0.0)/(alfa_m(0.0)+beta_m(0.0));
    nval = alfa_n(0.0)/(alfa_n(0.0)+beta_n(0.0));
    v_l = g_na*pow(mval,3)*hval*v_na+g_k*pow(nval,4)*v_k;
    v_l = -v_l/g_l;

/*  Step 13. - Run simulation */
    nstep = 0;
    getmem = 1;
    while ( nstep < maxstep ) {
        nstep++;
//        if ( nstep%1000 == 0 ) printf("\r Msec %d",nstep/1000);

/*  Phase 1. - Update Synaptic activity */
        for ( k=0 ; k<3*nodes-2 ; k++ ) lhs.a[k] = rhs.a[k] = 0.0;
        for ( k=0 ; k<nodes ; k++ ) {
            OldAmp[k] = NewAmp[k];
            NewAmp[k] = 0.0;
        }
        for ( k=0 ; k<cell->ndend ; k++ ) {
            bnew = cell->dendlist[k].root;
            Update_Synapses(bnew);
        }
        for ( k=0 ; k<3*nodes-2 ; k++ ) {
            lhs.a[k] += StoredLHS[k];
            rhs.a[k] += StoredRHS[k];
        }

/*  Phase 2. - Update HH channel variables */
        vs = v[nodes-1];
        aval = dt*alfa_h(vs);
        bval = dt*beta_h(vs);
        tmp = 0.5*(aval+bval);
        hval = (aval+(1.0-tmp)*hval)/(1.0+tmp);
        aval = dt*alfa_m(vs);
        bval = dt*beta_m(vs);
        tmp = 0.5*(aval+bval);
        mval = (aval+(1.0-tmp)*mval)/(1.0+tmp);
        aval = dt*alfa_n(vs);
        bval = dt*beta_n(vs);
        tmp = 0.5*(aval+bval);
        nval = (aval+(1.0-tmp)*nval)/(1.0+tmp);

/* Phase 3. - Compute somal conductance and contribution to current */
        gs = g_l;                       /* Leakage conductance */
        sc = g_l*v_l;                   /* Leakage contribution to somal current */
        tmp = g_na*hval*pow(mval,3);    /* Sodium conductance */
        gs += tmp;
        sc += tmp*v_na;                 /* Sodium contribution to somal current */
        tmp = g_k*pow(nval,4);          /* Potassium conductance */
        gs += tmp;
        sc += tmp*v_k;                  /* Potasium contribution to somal current */

/*  Phase 4. - Complete somal entry of LHS and RHS matrices */
        gs *= 0.5*dt;
        lhs.a[3*nodes-3] += gs;
        rhs.a[3*nodes-3] -= gs;

/*  Phase 5. - Step potential forward */
        Matrix_Vector_Multiply( &rhs, v, x);
        x[nodes-1] += sc*dt;
        tmp = 0.5*dt;
        for ( k=0 ; k<nodes ; k++ ) x[k] -= tmp*(OldAmp[k]+NewAmp[k]);
        cgs( &getmem, &lhs, x, v, CGS);

/* Phase 6. - Test for spikes */
        if ( nstep == 1 ) {
            vm2 = 0.0;
            vm1 = v[nodes-1];
        } else {
            vm0 = v[nodes-1];
            if ( !spk ) {
                spk = ( vm0 > 50.0 && vm1 > vm2 && vm1 > vm0 );
                if ( spk ) {
                    if ( nstep >= 1000*NT ) {
                        nspk++;
                        tnow = dt*((double) nstep)-1000.0;
                        tmp = tnow+0.5*dt*(vm2+3.0*vm0-4.0*vm1)/(vm0-2.0*vm1+vm2);
                        nn = ((int) floor(tmp));
                        if ( fmod(tmp,1.0)>0.5 ) nn++;
                        fp = fopen(OUTPUT,"a");
                        fprintf(fp,"%d\n",nn);
                        fclose(fp);
                    } else {
                        printf("\nUnrecorded spike fired\n");
                    }
                }
            }
            vm2 = vm1;
            vm1 = vm0;
        }

/*  Phase 7. - Reset spike flag */
        if ( vs < 0.0 && spk == 1 ) spk = 0;
        if ( nstep%(500*NT) == 0 ) {
            tnow = dt*((double) nstep/NT);
            printf("\rReached time %5.1lf \t Spikes so far %d", tnow, nspk);
        }
    }
    return 0;
}


 /*******************************************************
                 Function to output information
  *******************************************************/
void Output_Information(branch *b, FILE *fp)
{
    int k;
    double pos;
    synapse *syn;

    if ( b->child ) Output_Information( b->child, fp);
    if ( b->peer ) Output_Information( b->peer, fp);

    syn = b->synlist;
    while ( syn ) {
        pos = (syn->xp)/(b->hseg);
        fprintf(fp," %12.6lf \t %d \n",pos, syn->SynTime);
        syn = syn->NextSyn;
    }
    fprintf(fp,"End of branch\n");
    return;
}


 /******************************************************
        Function to constuct sparse matrices
  ******************************************************/
void Generate_Dendrite( branch *b, int *counter)
{
    int k, nc, CurrentNode;
    extern double pi;
    extern SparseMatrix lhs, rhs;
    branch *btmp;
    double SumL, SumR;

/* Step 1 - Recurse to the end of the dendrite */
    if ( b->child ) Generate_Dendrite( b->child, counter);
    if ( b->peer ) Generate_Dendrite( b->peer, counter);

/* Step 2 - Fill in matrix entries for a branch point node */
    nc = b->nc;
    CurrentNode = (b->first)-(nc-1);
    if ( b->child ) {
        btmp = b->child;
        SumR = SumL = 0.0;
        while ( btmp ) {
            lhs.a[*counter] = 0.0;
            rhs.a[*counter] = -0.25*pi*pow(btmp->diam,2)/(btmp->hseg);
            SumL += 0.5*pi*(btmp->diam)*(btmp->hseg);
            SumR += 0.25*pi*pow(btmp->diam,2)/(btmp->hseg);
            lhs.col[*counter] = rhs.col[*counter] = btmp->first;
            (*counter)++;
            btmp = btmp->peer;
        }
        lhs.a[*counter] = SumL+0.5*pi*(b->diam)*(b->hseg);
        rhs.a[*counter] = SumR+0.25*pi*pow(b->diam,2)/(b->hseg);
        lhs.col[*counter] = rhs.col[*counter] = CurrentNode;
        (*counter)++;
        lhs.a[*counter] = 0.0;
        rhs.a[*counter] = -0.25*pi*pow(b->diam,2)/(b->hseg);
        if ( CurrentNode == b->first ) {
            lhs.col[*counter] = rhs.col[*counter] = b->junct;
        } else {
            lhs.col[*counter] = rhs.col[*counter] = CurrentNode+1;
        }
        (*counter)++;
        lhs.StartRow[CurrentNode+1] = rhs.StartRow[CurrentNode+1] = *counter;
    } else {

/* Step 3 - Fill in matrix entries for a terminal node */
        lhs.a[*counter] = 0.5*pi*(b->diam)*(b->hseg);
        rhs.a[*counter] = 0.25*pi*pow(b->diam,2)/(b->hseg);
        lhs.col[*counter] = rhs.col[*counter] = CurrentNode;
        (*counter)++;
        lhs.a[*counter] = 0.0;
        rhs.a[*counter] = -0.25*pi*pow(b->diam,2)/(b->hseg);
        if ( CurrentNode == b->first ) {
            lhs.col[*counter] = rhs.col[*counter] = b->junct;
        } else {
            lhs.col[*counter] = rhs.col[*counter] = CurrentNode+1;
        }
        (*counter)++;
        lhs.StartRow[CurrentNode+1] = rhs.StartRow[CurrentNode+1] = *counter;
    }

/* Step 4 - Fill in matrix entries for an internal node */
    for ( k=nc-1 ; k>0 ; k-- ) {
        CurrentNode++;
        lhs.a[*counter] = 0.0;
        rhs.a[*counter] = -0.25*pi*pow(b->diam,2)/(b->hseg);
        lhs.col[*counter] = rhs.col[*counter] = CurrentNode-1;
        (*counter)++;
        lhs.a[*counter] = pi*(b->diam)*(b->hseg);
        rhs.a[*counter] = 0.5*pi*pow(b->diam,2)/(b->hseg);
        lhs.col[*counter] = rhs.col[*counter] = CurrentNode;
        (*counter)++;
        lhs.a[*counter] = 0.0;
        rhs.a[*counter] = -0.25*pi*pow(b->diam,2)/(b->hseg);
        if ( CurrentNode == b->first ) {
            lhs.col[*counter] = rhs.col[*counter] = b->junct;
        } else {
            lhs.col[*counter] = rhs.col[*counter] = CurrentNode+1;
        }
        (*counter)++;
        lhs.StartRow[CurrentNode+1] = rhs.StartRow[CurrentNode+1] = *counter;
    }
    return;
}


 /*************************************************************
          Function to remove a branch from a branch list
  *************************************************************/
void Remove_Branch(branch **head, branch *b)
{
    if ( *head == NULL || b == NULL ) return;
    if ( *head == b ) {
        *head = b->child;
        if ( *head != NULL )  (*head)->parent = NULL;
    } else {
        b->parent->child = b->child;
        if ( b->child != NULL ) b->child->parent = b->parent;
    }
    b->parent = NULL;
    b->child = NULL;
    return;
}


 /*********************************************
      Function to count synapses on branches
  *********************************************/
int Count_Synapses( branch *bstart, branch *bnow)
{
    static int n;
    synapse *syn;

    if ( bstart == bnow ) n = 0;
    if ( bnow != NULL ) {
        if ( bnow->child ) Count_Synapses(bstart, bnow->child);
        if ( bnow->peer ) Count_Synapses(bstart, bnow->peer);
        syn = bnow->synlist;
        while ( syn ) {
            n++;
            syn = syn->NextSyn;
        }
    }
    return n;
}


 /*******************************************************
             Function to count number of branches
  *******************************************************/
int Count_Branches( branch *bstart, branch *bnow)
{
    static int n;

    if ( bstart == bnow ) n = 0;
    if ( bnow ) {
        if ( bnow->child ) Count_Branches(bstart, bnow->child);
        if ( bnow->peer ) Count_Branches(bstart, bnow->peer);
        n++;
    }
    return n;
}


 /*******************************************************
       Function to enumerate the nodes on a dendrite
  *******************************************************/
void Enumerate_Nodes(branch *bnow, int *FirstNode )
{
    branch *btmp;

    if ( bnow->child ) Enumerate_Nodes( bnow->child, FirstNode );
    if ( bnow->peer ) Enumerate_Nodes( bnow->peer, FirstNode );

    btmp = bnow->child;
    while ( btmp ) {
        btmp->junct = *FirstNode;
        btmp = btmp->peer;
    }
    *FirstNode += bnow->nc;
    bnow->first = *FirstNode-1;
    return;
}


 /*************************************************************
        Function to build a test dendrite from its root
  *************************************************************/
void Build_Test_Dendrite( branch **head, branch *root)
{
    double tmp;
    branch *bnow, *bnext, *btmp;

    bnow = *head;
    while ( bnow != NULL ) {

/*  Store bnow's child in case it's corrupted */
        bnext = bnow->child;

/*  Decide if proximal end of bnow is connected to distal end of root */
        tmp = pow(bnow->xl-root->xr,2)+
              pow(bnow->yl-root->yr,2)+
              pow(bnow->zl-root->zr,2);
        if ( tmp <= 0.01 ) {

/*  Remove bnow from the branch list */
            Remove_Branch( head, bnow);

/*  Connect bnow to the root as the child or a peer of the child.
    Initialise childs' children and peers to NULL as default */
            bnow->child = NULL;
            bnow->peer = NULL;
            bnow->parent = root;

/*  Inform root about its child if it's the first child, or add
    new child to first child's peer list */
            if ( root->child != NULL ) {
                btmp = root->child;
                while ( btmp->peer != NULL ) btmp = btmp->peer;
                btmp->peer = bnow;
            } else {
                root->child = bnow;
            }
        }

/*  Initialise bnow to next branch in list */
        bnow = bnext;
    }

/* Iterate through remaining tree */
    if ( root->child ) Build_Test_Dendrite( head, root->child);
    if ( root->peer ) Build_Test_Dendrite( head, root->peer);
    return;
}


 /**********************************************************
           Function to initialist synaptic input to dendrite
  **********************************************************/
void Initialise_Synapses( branch *b)
{
    extern unsigned int ix, iy, iz;
    extern SparseMatrix lhs, rhs;
    int k;
    double ratio, interval;
    synapse *syn;

    if ( b->child ) Initialise_Synapses( b->child );
    if ( b->peer ) Initialise_Synapses( b->peer );

    syn = b->synlist;
    while ( syn ) {
        ratio = (syn->xp)/(b->hseg);
        if ( fmod(ratio,1.0) >= 0.5 ) {
            k = ((int) ceil(ratio));
        } else {
            k = ((int) floor(ratio));
        }
        if ( k == 0 ) {
            syn->cn = b->junct;
        } else {
            syn->cn = b->first-k+1;
        }
        k = lhs.StartRow[syn->cn];
        while ( lhs.col[k] != syn->cn ) k++;
        syn->diag = k;
        syn->gval = 0.0;
        interval = -(((double) NT*1000)/RATE)*log(ran(&ix, &iy, &iz));
        if ( fmod(interval,1.0) <= 0.5 ) {
            syn->SynTime = -((int) floor(interval));
        } else {
            syn->SynTime = -((int) ceil(interval));
        }
        syn = syn->NextSyn;
    }
    return;
}


 /**********************************************************
           Function to update synaptic input to dendrite
  **********************************************************/
void Update_Synapses( branch *b)
{
    extern unsigned int ix, iy, iz;
    extern double dt, *NewAmp;
    extern SparseMatrix lhs, rhs;
    int k;
    double interval, tmp, gnew, gold;
    synapse *syn;

    if ( b->child ) Update_Synapses( b->child);
    if ( b->peer ) Update_Synapses( b->peer);

    syn = b->synlist;
    while ( syn ) {
        gold = syn->gval;
        (syn->SynTime)++;
        if ( syn->SynTime < 1 ) {
            syn->gval = 0.0;
        } else if ( syn->SynTime < syn->MaxStep ) {
            syn->gval = syn->SynCond->g[syn->SynTime];
        } else {
            interval = -(((double) NT*1000)/RATE)*log(ran(&ix, &iy, &iz));
            syn->gval = 0.0;
            if ( fmod(interval,1.0) <= 0.5 ) {
                syn->SynTime = -((int) floor(interval));
            } else {
                syn->SynTime = -((int) ceil(interval));
            }
        }
        gnew = syn->gval;
        tmp = 0.5*dt;
        lhs.a[syn->diag] += tmp*gnew;
        rhs.a[syn->diag] -= tmp*gold;
        NewAmp[syn->cn] -= gnew*(syn->vsyn);
        syn = syn->NextSyn;
    }
    return;
}


 /*********************************************************
            Function to assign branch nodes
  *********************************************************/
void Assign_Branch_Nodes( branch *b, double *h )
{
    b->nc = ((int) ceil((b->plen)/(*h)));     /* nc is the number of compartments in branch */
    b->hseg = (b->plen)/((double) b->nc);     /* hseg is the compartment length */

    if ( b->child ) Assign_Branch_Nodes( b->child, h);
    if ( b->peer ) Assign_Branch_Nodes( b->peer, h);
}


 /************************************************************
         Function returns primitive uniform random number.
  ************************************************************/
double ran(unsigned int *ix, unsigned int *iy, unsigned int *iz)
{
    double tmp;

/*  1st item of modular arithmetic  */
    *ix = (171*(*ix))%30269;
/*  2nd item of modular arithmetic  */
    *iy = (172*(*iy))%30307;
/*  3rd item of modular arithmetic  */
    *iz = (170*(*iz))%30323;
/*  Generate random number in (0,1)  */
    tmp = ((double) (*ix))/30269.0+((double) (*iy))/30307.0
          +((double) (*iz))/30323.0;
    return fmod(tmp,1.0);
}


 /**********************************************************
     Multiplies sparse matrix a[ ][ ] with vector v[ ]
  **********************************************************/
void Matrix_Vector_Multiply( SparseMatrix *a, double *v , double *b)
{
    int i, j, k, n;

    n = a->n;
    for ( j=0 ; j<n ; j++) {
        k = a->StartRow[j+1];
        for( b[j]=0.0,i=(a->StartRow[j]) ; i<k ; i++ ) {
            b[j] += (a->a[i])*v[a->col[i]];
        }
    }
    return;
}


 /***********************************************
        Allocate memory to a sparse matrix
  ***********************************************/
void Matrix_Malloc( SparseMatrix *a, int n, int w)
{
    a->a = (double *) malloc( w*sizeof(double) );
    a->col = (int *) malloc( w*sizeof(int) );
    a->StartRow = (int *) malloc( (n+1)*sizeof(int) );
    a->n = n;
    return;
}


 /**********************************************
     De-allocates memory of a sparse matrix
  **********************************************/
void Matrix_Free( SparseMatrix *a)
{
    free(a->a);
    free(a->col);
    free(a->StartRow);
    free(a);
}


void cgs(int *getmem, SparseMatrix *a, double *b, double *x, double tol)
{
    long int i, k, n, repeat;
    static int start=1;
    double rho_old, rho_new, alpha, beta, sigma, err;
    static double *p, *q, *r, *u, *v, *rb, *y;

/* Step 1 - Check memory status */
    n = a->n;
    if ( start ) {
        *getmem = 1;
        start = 0;
    }
    if ( *getmem ) {
        if ( p ) free(p);
        p = (double *) malloc( n*sizeof(double) );
        if ( q ) free(q);
        q = (double *) malloc( n*sizeof(double) );
        if ( r ) free(r);
        r = (double *) malloc( n*sizeof(double) );
        if ( u ) free(u);
        u = (double *) malloc( n*sizeof(double) );
        if ( v ) free(v);
        v = (double *) malloc( n*sizeof(double) );
        if ( rb ) free(rb);
        rb = (double *) malloc( n*sizeof(double) );
        if ( y ) free(y);
        y = (double *) malloc( n*sizeof(double) );
        *getmem = 0;
    }

/* Step 2 - Initialise residual, p[ ] and q[ ] */
    Matrix_Vector_Multiply( a, x, r);
    for ( rho_old=0.0,i=0 ; i<n ; i++ ) {
        r[i] = b[i]-r[i];
        rho_old += r[i]*r[i];
        rb[i] = r[i];
        p[i] = 0.0;
        q[i] = 0.0;
    }
    if ( rho_old<tol*((double) n) ) return;

/* The main loop */
    rho_old = 1.0;
    do {

/* Compute scale parameter for solution update */
        for ( rho_new=0.0,i=0 ; i<n ; i++ ) rho_new += r[i]*rb[i];
        beta = rho_new/rho_old;

/* Update u[ ] and p[ ] */
        for ( i=0 ; i<n ; i++ ) {
            u[i] = r[i]+beta*q[i];
            p[i] = u[i]+beta*(q[i]+beta*p[i]);
        }

/* Update v[ ] and compute sigma */
        Matrix_Vector_Multiply( a, p, v);
        for ( sigma=0.0,i=0 ; i<n ; i++ ) sigma += rb[i]*v[i];

/* Compute alpha and update q[ ], v[ ] and x[ ] */
        if ( sigma == 0.0 ) {
            printf(" Trouble ");
            for ( i=0 ; i<n ; i++ ) {
                printf("\n%20.16lf",v[i]);
                getchar( );
            }
        }
        alpha = rho_new/sigma;
        for ( i=0 ; i<n ; i++ ) {
            q[i] = u[i]-alpha*v[i];
            v[i] = alpha*(u[i]+q[i]);
            x[i] += v[i];
        }

 /* Update r[ ] and estimate error */
        Matrix_Vector_Multiply( a, v, y);
        for ( err=0.0,i=0 ; i<n ; i++ ) {
            r[i] -= y[i];
            err += r[i]*r[i];
        }
        rho_old = rho_new;
              if ( rho_old == 0.0 ) {
            printf(" Trouble rho_old ");
            for ( i=0 ; i<n ; i++ ) {
                printf("\n%20.16lf",v[i]);
                getchar( );
            }
        }
        repeat = ( err > tol*((double) n) );
    } while ( repeat );
    return;
}


 /********************************************************************
                    ALPHA for ACTIVATION OF SODIUM
  *******************************************************************/
double alfa_m( double volt )
{
    double tmp;
    static double fac;
    static int start=1;

    if ( start ) {
        fac = pow(3.0,0.1*CELSIUS-0.63);
        start = !start;
    }
    tmp = -0.1*(volt-25.0);
    if ( fabs(tmp)<0.001 ) {
        tmp = 1.0/(((tmp/24.0+1.0/6.0)*tmp+0.5)*tmp+1.0);
    } else {
        tmp = tmp/(exp(tmp)-1.0);
    }
    return tmp*fac;
}


 /********************************************************************
                    BETA for ACTIVATION OF SODIUM
  ********************************************************************/
double beta_m( double volt )
{
    double tmp;
    static double fac;
    static int start=1;

    if ( start ) {
        fac = pow(3.0,0.1*CELSIUS-0.63);
        start = !start;
    }
    tmp = volt/18.0;
    return 4.0*fac*exp(-tmp);
}


 /********************************************************************
                    ALPHA for INACTIVATION OF SODIUM
  ********************************************************************/
double alfa_h( double volt )
{
    double tmp;
    static double fac;
    static int start=1;

    if ( start ) {
        fac = pow(3.0,0.1*CELSIUS-0.63);
        start = !start;
    }
    tmp = 0.05*volt;
    return 0.07*fac*exp(-tmp);
}


 /********************************************************************
                    BETA for INACTIVATION OF SODIUM
  ********************************************************************/
double beta_h( double volt )
{
    double tmp;
    static double fac;
    static int start=1;

    if ( start ) {
        fac = pow(3.0,0.1*CELSIUS-0.63);
        start = !start;
    }
    tmp = -0.1*(volt-30.0);
    return fac/(exp(tmp)+1.0);
}


 /********************************************************************
                    ALPHA for ACTIVATION OF POTASSIUM
  ********************************************************************/
double alfa_n( double volt )
{
    double tmp;
    static double fac;
    static int start=1;

    if ( start ) {
        fac = pow(3.0,0.1*CELSIUS-0.63);
        start = !start;
    }
    tmp = -0.1*(volt-10.0);
    if ( fabs(tmp)<0.001 ) {
        tmp = 0.1/(((tmp/24.0+1.0/6.0)*tmp+0.5)*tmp+1.0);
    } else {
        tmp = 0.1*tmp/(exp(tmp)-1.0);
    }
    return tmp*fac;
}



 /********************************************************************
                    BETA for ACTIVATION OF POTASSIUM
  ********************************************************************/
double beta_n( double volt )
{
    double tmp;
    static double fac;
    static int start=1;

    if ( start ) {
        fac = pow(3.0,0.1*CELSIUS-0.63);
        start = !start;
    }
    tmp = 0.0125*volt;
    return 0.125*fac*exp(-tmp);
}


 /********************************************
        Computes a conductance profile
  ********************************************/
cond *ConductanceProfile( double dt,   /* Integration time step */
                          double tau,  /* Synaptic time constant */
                          double tol,  /* Determines off-threshold for synapse */
                          double gmax  /* Maximum conductance (mS) */ )
{
    int k;
    cond *out;
    double tmp, told, tnew;

    out = (cond *) malloc( sizeof(cond) );
    out->dt = dt;
    out->tau = tau;
    out->tol = tol;
    out->gmax = gmax;

/* Iterate to find duration of pulse */
    tmp = 1.0-log(tol);
    tnew = tmp;
    do {
        told = tnew;
        tnew = tmp+log(told);
    } while ( fabs(tnew-told)>5.e-11 );
    out->n = ((int) ceil(tau*tnew/dt));
    out->g = (double *) malloc( (out->n)*sizeof(double) );
    out->g[0] = 0.0;
    for ( k=1 ; k<(out->n) ; k++ ) {
        tmp = dt*((double) k)/tau;
        out->g[k] = gmax*tmp*exp(1.0-tmp);
    }
    return out;
}

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