Remarks on the simulation

1

St

perc.

0 time T

Figure 11.1: The HBMc. The first passage times(H_i)(green) of a Brownian motion (black, panel A) with drift ±ν₀ at borders±b indicate the times of the percept changes (orange, panel B). The Brownian motion is assumed to summarize the activity difference of two conflicting neuronal populations with only two parameters. Additionally, for a given t the value oft^∗ is visualized.

11.1.2 Effect of single parameter changes

The influence of bandν0 on the mean dominance time µand the CV is shown in Figure 11.2.

Note thatbhas opposite effects on the mean (2b/ν₀) and the CV= 1/√

2bν₀ of the dominance times, whereasν₀ has the same effect on mean and CV as it is in the denominator of both quantities (recall Proposition 8.6 for the transformation ofband ν0 toµand CV).

● ●

A

⇒ b

µ

CV

⇒

B

ν0⇒ µ CV

Figure 11.2: Influence of the HBMc parameters on mean and CV of the domi-nance times. The influence of increasing the two parameters b (panel A) and ν₀ (panel B) while leaving the other one constant on µ and CV=σ/µ is shown.

11.1.3 Remarks on the simulation

The simulation of the Hierarchical Brownian model (for continuous stimulation) on the interval [0, T] mainly relies on the simulation of a Brownian motion on [0, T]. One way to generate such a stochastic process is to use the discrete skeleton 0 =t0 < t1 < . . . < t_{T /∆}=T with step width ti−ti−1 =: ∆>0 ∀i= 1, . . . , T /∆ by simulating i.i.d. random variables I₁^∆, I₂^∆, . . . , I_{T /∆}^∆ with normal distribution with mean 0 and variance ∆ (e.g., Asmussen and Glynn, 2007). To simulate the perception process (P_t^∆)t=0,∆,2∆,...during continuous presentation on the discrete skeleton with step width ∆, we use fori≥1 the summing of the increments as follows

11. A hierarchical Brownian motion model

P_t^∆_i =P_t^∆_i−1+I_i^∆+ ∆·

(ν₀, if argmax_ti{P_t^∆

i :P_t^∆_i ≤ −b}>argmax_ti{P_t^∆

i :P_t^∆_i ≥b}, (−ν₀), if argmax_ti{P_t^∆

i :P_t^∆_i ≤ −b}<argmax_ti{P_t^∆

i :P_t^∆_i ≥b}, withP₀^∆:=−b and argmax{}:=−1.

Hence, in every simulation step we have to check whether the perception process is aboveb or below−b. Implementing this in afor()-loop is in the standard statistical programming package R time consuming in particular for small step widths ∆. Therefore, we propose to outsource the simulation toC++which is much more efficient while performing standard loops. Nevertheless, the normal distributed random variables are simulated in R and then transferred as function input to theC++-function. We compare the runtimes using the two different programming languages in Figure 11.3 for three different time horizonsT and four different step widths ∆. The dependence of the computation time on the time horizonT and the programming language –C++(yellow points) leads to faster simulations – is clearly visible.

The choice of the step width ∆ is another crucial point in the simulation. Large ∆ leads to inaccurate simulation, whereas a small ∆ increases the computational effort (compare the different symbols in Figure 11.3). We recommend the step width ∆ = 0.01.

● ●

●

●

●

●

●

●

4 7 10

−5 0 5 10

log(T) [s]

log(comp. time) [s]

●

∆=

0.001 0.01 0.1 1

language R C++

Figure 11.3: Comparison of the computation times for simulation of the HBMc.

Mean computation times (in ten trials) with R are printed red, whereas the C++ results are printed yellow. The simulation of the HBMc is performed using different time horizonsT and step widths∆ visualized by different symbols (see legend). Both axes are logarithmic due to the different magnitudes.

11.2 Intermittent presentation

In this section we introduce the Hierarchical Brownian Motion model for intermittent pre-sentation (Section 11.2.1), discuss its assumptions and its relation to the HMM (Sections

128

11. A hierarchical Brownian motion model

11.2.2 and 11.2.3) and elaborate on the effects of single parameter changes (Section 11.2.4).

Moreover, in Sections 11.2.5 and 11.2.6, the dominance time distributions dependent on the following hidden state and Markov properties are examined. Finally, hints for simulation are given (Section 11.2.7).

11.2.1 The model

In the Hierarchical Brownian Motion model for intermittent presentation (HBMi), we require mechanisms for long dominance times in the stable state as well as for short dominance times in the unstable state. In order to describe the responses to intermittent and continuous presentation in one model framework, we assume the identical perceptual process as in the HBMc (Section 11.1) during phases of stimulus presentation. The periods of blank display represent the only difference in the experimental setup to continuous presentation. In these periods, we assume additional neuronal mechanisms. In particular, we assume that the perceptual process then takes on one of two mean drifts,ν_S in the stable state andν_U ≥ν_S in the unstable state, with potentially opposite signs of ν0 and νS for increased stability (Figure 11.4). Note that the driftsν_S andν_U are not necessarily constant across the whole period of blank display, but they denote the mean drift of the process, which is sufficient to describe the distribution of dominance times. Interestingly, additional assumptions on the temporal behavior of the drift terms could also allow describing the impact of the lengths of blank displays (compare Section 15.2). Further, in the unstable state the border b_U at which perception and drift direction change is assumed smaller than the borderbS during stable perception. Switches between the stable and unstable state will be caused by a similar mechanism in a so-called background processB described later in this section.

Within a state (S or U), the fluctuation of the perception process between the borders is assumed analogous to the HBMc, except that the borders are dependent on the hidden state and that the drift isν₀ during presentation and ν_S or ν_U during blank display. Formally, we denote by PR and BL the sets of all periods of stimulus presentation and blank display, respectively. Assuming that we start a trial with a presentation interval and then switch regularly between presentation intervals of lengthl_p and blank display of lengthl_b, PR and BL are given by

PR =

T /(lp+lb)

[

i=1

[(i−1)(lp+l_b); (i−1)(lp+l_b) +lp)

BL =

T /(lp+lb)

[

i=1

[(i−1)(l_p+l_b) +l_p;i(l_p+l_b)) as shown in Figure 11.4.

The perception processP := (P_t)_t is then given by dP_t=

(Stν0dt+dWt, ift∈PR, S_tν_Y˜

tdt+dW_t, ift∈BL,

where ˜Y_t ∈ {S, U} denotes the hidden state at time t ≥ 0 and (W_t)_t denotes a standard Brownian motion. As a result, the mean drift per second is given by the weighted mean

ν_S^∗ := l_b·ν_S+l_p·ν₀ lb+lp

and ν_U^∗ := l_b·ν_U+l_p·ν₀ lb+lp

(11.2)

11. A hierarchical Brownian motion model

A

drift

νS

ν0

νU

l_pl_b

P

−b_S

−b_U 0 b_U b_S

...

L R

perc. drift

B

νS

ν0

νU

l_pl_b

P

−b_S

−b_U 0 b_U b_S

...

L R

perc.

BL PR

Figure 11.4: The perception process P in the HBMi during intermittent presen-tation. During presentation, P has driftν0. During blank displays (yellow), P has drift νS

in the stable phase (A), and drift νU in the unstable phase (B). Typically, we have νS ≤ν0

and ν_U ≥ν₀. The borders are b_S (light blue horizontal line) in the stable state and b_U (blue horizontal line) in the unstable state.

for statesS andU, respectively. Because the periods l_b and lp are typically short in relation to a dominance time, the behavior of P can be approximated by a Brownian motion with absolute driftsν_S^∗ and ν_U^∗, respectively. As in the HBMc, the sign of the drift St:=S(Pt, t) changes at every first hitting time of the respective border, i.e.,

S(P_t, t) :=−sgn(P_t^∗), wheret^∗ :=t^∗(t) := sup{x:x < t,|P_x|=b_Yx} witht^∗(0) = 0.

We initialize P₀ = −b_Y˜

0 for the initial state ˜Y₀ which is the stable state with probability π^∗_start,S:=P( ˜Y₀=S). The perception then takes the valueL ifS_t= 1 andR ifS_t=−1 and switches at the first-hitting times (H_i)_i of the borders ±b_i comparable to equation (11.1).

Note that perception also changes during blank display. The dominance times are therefore again given byd_i :=H_i−Hi−1, i= 1,2, . . ..

In order to describe the switching between the two states S and U, we use an analogous upper hierarchical level with another pair of conflicting neuronal populations. Their difference activity is described by a so-called background process B := (B_t)_t (Figure 11.5 A, middle panel). B is also assumed to be a Brownian motion with drift. Its drift is assumed to vanish during presentation and to take the value±ν_B during blank display, where the sign of drift depends on the hidden state as follows

dBt=







dW˜_t, ift∈PR,

ν_Bdt+dW˜_t, ift∈BL,Y˜_t=S,

−ν_Bdt+dW˜t, ift∈BL,Y˜t=U,

(11.3)

where ( ˜W_t)_t is a Brownian motion independent of (W_t)_t. Again, the mean drift across PR and BL intervals isν_B^∗:= _l^lb·νB

b+lp.

The background processB evokes changes between the stable and the unstable state. Specif-ically, at the time of a percept change t^∗, the question of whether the process stays in the former state (S or U) or switches to the other state depends only on the value of B. Two borders, ˜bS and ˜bU, determine this switching as follows (see Figure 11.5). If the former state is S, the process remains stable if and only if Bt^∗ ≥˜b_S (first light blue arrow in panel A), while switching to the unstable state ifB_t^∗ <˜b_S (blue arrow, panel A). Analogously, if the

130

11. A hierarchical Brownian motion model

former state isU, the process switches to S if and only ifB_t^∗ ≥˜b_U (right light blue arrow, panel A), while staying inU if Bt^∗ <˜bU (blue arrow, panel B). After the percept change, the background processB is reset to zero and then follows its usual dynamic (eq. (11.3)), i.e., the sign of its drift changes if and only if the state has changed. Finally, as the perception process P fluctuates between ±b_S in the stable state and between±b_U in the unstable state, the value ofP is reset when the state changes, to the value sgn(P)b_S when changing to the stable state and to sgn(P)b_U when changing to the unstable state.

A

P

−b_S b_S

b_U

S U S

Bb~

S

b~

U

0 500

L R

time [s]

zoom

perc.

● ●

Index

B

P 0 b_U

U S

B 0 b~

U

408 458

L R

time [s]

perc.

Figure 11.5: The HBMi. The perception process P, the background process B and the resulting percept. (A): A simulation on [0,500]. (B): The same realization, zoomed in on the time interval [408,458]. Stable phases indicated by light gray background, unstable phases indicated by dark gray background. The beginnings of stable and unstable phases are marked with light blue and blue arrows, respectively.

Detailed model definition

Here, we state a precise mathematical model definition of the HBMi (as supplement to the prose definition given above).

11. A hierarchical Brownian motion model

Model 11.1. Hierarchical Brownian model in intermittent presentation (HBMi) LetbS>0,bU ∈(0, bS],˜bS >0and˜bU ∈Rbe four borders, letν0 >0, νS ∈R, νU ≥νS, νB>0 be drift parameters, T >0 be a time horizon, l_p >0 be the length of stimulus presentation and l_b >0 be the length of blank display. Furthermore, 0≤π_start,S^∗ ≤1 is the initial weight of the stable state.

Let(Wt)tand( ˜Wt)tbe independent standard Brownian motions andP₀^S be a Bernoulli(π_start,S^∗ )-distributed random variable. Then, the dynamics of the perception process P := (P_t)_t∈[0,T] and the background process B := (Bt)t∈[0,T] are assumed to be

dPt=







S(P_t, t)ν(P_t, B_t, t)dt+dW_t, if db_t= 0,

|b_S−bU|, if dbt6= 0∧Pt≤min(sgn(Pt)bS,sgn(Pt)bU),

−|b_S−b_U|, if db_t6= 0∧P_t>min(sgn(P_t)b_S,sgn(P_t)b_U), with P₀=−b₀, (b_t) the border process defined below and the process(S_t) depending on the sign of the last hit border ±b_t

S_t:=S(P_t, b_t, t) :=−sgn(P_t^∗), where t^∗:=t^∗(t) := sup{x:x < t,|P_x|=b_x} witht^∗(0) = 0.

The drift ν(Pt, Bt, t) depends on the background process (see below) ν(Pt, Bt, t) :=

(ν₀, if t∈PR, ν_t^blank, if t∈BL, where ν_t^blank:=ν^blank(Pt, Bt, t)

ν_t^blank(Pt, Bt, t) :=







V₀, ift^∗ = 0,

νS, ift^∗ >0∧((Bt^∗ ≥˜bS∧ν_t^blank∗ =νS) or (Bt^∗ ≥˜bU∧ν_t^blank∗ =νU)), νU, ift^∗ >0∧((Bt^∗ <˜bS∧ν_t^blank∗ =νS) or (Bt^∗ <˜bU∧ν_t^blank∗ =νU)), whereV0 is vS ifP₀^S = 1and vU otherwise, and the border process(bt) :=b(ν_t^blank, t)is assumed to be

b(ν_t^blank, t) :=

(bS, ifν_t^blank=νS, b_U, ifν_t^blank=ν_U.

The perception at time t≥0 takes the value L if S_t = 1 and R if S_t= −1 and (H_i)_i=0,1,...

denotes the process of first passage times as defined in (11.1) with b replaced byb_t. For the background process(Bt)t∈[0,T] we assume B0 = 0 and

dB_t=

(1t∈BLS(B˜ t, Pt, t)νBdt+dW˜t, if dSt= 0,

−sgn(B_t)Bt, if dSt6= 0, with the sign of drift

S˜_t:= ˜S(ν_t^blank, t) :=

(1, if ν_t^blank=νS,

−1, if ν_t^blank=ν_U.

We say that the HBMi is in stateS at timetifv_t^blank equalsν_S and in stateU otherwise.

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11. A hierarchical Brownian motion model

11.2.2 Discussion of assumptions and interpretation of model parameters

Im Dokument Stochastic models for variability changes in neuronal point processes (Seite 135-141)