# Impulse Pattern Formulation (IPF) Brain Model

Rolf Bader  Institute of Musicology
University of Hamburg
Neue Rabenstr. 13, 20354 Hamburg, Germany

###### Abstract

A new brain model is introduced, based on the Impulse Pattern Formulation
(IPF) already established for modeling and understanding musical instrument
and rhythm perception and production. It assumes the brain works with
impulses, neural bursts, ejected from an arbitrary reference point in the
brain, arriving at other reflecting brain regions, and returning to the
reference point delayed and damped. A plasticity model is suggested to adjust
reflection strength in time. The model is systematically studied with 50
reflection points by varying the amount of excitatory vs. inhibitory neurons,
the presence or absence of plasticity or external sensory input, and the
strength of the input and plasticity in terms of system adaptation to an input
or to the system itself. The Brain IPF shows adaptation to an external
stimulus, which is stronger without plasticity, showing the active brain not
being a simple passive _tabula rasa_. A relation of 10-20% of inhibitory vs.
excitatory neurons, as found in the brain, shows a maximum adaptation to an
external stimulus compared to all other relations, pointing to an optimum of
this relation concerning adaptation. Although the model has no fixed
timescale, when assuming strong brain periodicities only up to about 100 Hz,
the reflection strength of the model is highest for delays of around 300 ms,
corresponding to Event-Related Potential (ERP) timescales of brain potentials
most often found roughly between 100 - 400 ms. The mean convergence times of
the model correspond to short-time memory time scales with a mean of five
seconds for converging IPFs. The Brain IPF is computationally very cheap,
highly flexible, and with musical instruments already found to be of high
predictive precision. Therefore, in future studies, the Brain IPF might be a
model able to understand very large systems composed of an ensemble of brains
as well as cultural artifacts and ecological entities.

##  I Introduction

Existing brain models vary in complexity, scaling, and mathematical
modeling26. On a low level, models solving differential equations of single
neurons have been proposed like the Hodgkin-Huxley23 or the Izhikevich
model25, where, e.g., coincidence detection of spikes leaving the cochlear
have been modelled6. Dynamical networks have also been proposed, modeling
music large-scale forms using a FitzHugh-Nagumo (FHN) model40, based on the
empirical finding using EEG measurements of synchronization of brain parts
with increasing musical tension22. The transition of mechanical waves on the
basilar membrane into neural spikes has shown phase synchronization between
tone partials already on this low level7, which is already able to reproduce
timbre features like a bifurcating spectrum in a so-called surface tone of a
cello5. These methods can be considered bottom-up models as they try to
understand the brain by modeling coupled single neurons.

Top-down models using global working principles of the brain have also been
proposed. Friston assumes the brain to work according to a free-energy
principle, in analogy to thermodynamics, as a change of equilibrium and
surprises, which is also applied to hearing and music17. Haken uses methods of
Synergetics to calculate Gestalt perception21 analytically. Baars assumes the
brain works as a global workspace, where brain parts synchronize and de-
synchronize1. The appearance and role of consciousness are also discussed
within this framework. Such synchronization is also found with expectation12
and musical tension2240.

Neural networks have also been proposed as connectionist19, 20 or as self-
organizing28 networks. Musical applications in the connectionist sense concern
music analysis18 or composition11. They have often proven to be capable of
analyzing and synthesizing music realistically. Still, their disadvantage is
the impossibility of deriving the reason for successful modeling from the
learned neural weights. Furthermore, they often need large databases to
succeed. On the other hand, self-organizing maps (SOM) allow deriving reasons
for the classification and sorting of input data in terms of music theory15,
defining ethnic groups3, or in musical acoustics39. These methods are
increasingly used in Computational Phonogram Archiving4 classifying, sorting,
or analyzing big databases in ethnomusicology10, streaming platforms, or all
kinds of audio archives9.

Consciousness and conscious content are found to be spatiotemporal patterns as
found with olfactory perception29 or auditory perception3837. The brain shows
phases of periodicity interrupted by chaotic transient phases within
milliseconds. The periodic phases show repeating activity patterns of neural
clusters forming structures in space and time. Different conscious content is
thereby represented by different spatiotemporal patterns.

Enlarging brain dynamics into society was proposed by Freeman as the idea of a
society of brains16 as a course of culture. Using the IPF and enlarging this
framework by including cultural artifacts has been proposed in a Physical
Culture Theory2.

Within this line of reasoning, the present paper proposes a brain model based
on the Impulse Pattern Formulation (IPF). It has first been proposed as a
general method for understanding and modeling musical instruments 8. As a
general method, it can compare musical instrument families, like plucked,
blown, struck, or bowed instruments3334. The method assumes musical
instruments to work with impulses, short energy bursts that are produced at a
reference point, travel through the instrument, are reflected at one or
several instrument parts, and return as a damped impulse to the reference
point, which reacts to this impulse by sending a new impulse. To arrive at a
most general model, the shape of the impulses is not considered and can be
inserted into the resulting impulse pattern later.

As a result of multiple send-outs and reflections, an impulse pattern results.
This pattern is represented by a system variable g which is updated at each
iteration time and interpreted as a time interval or amplitude. So a constant
value of g is a steady periodicity, corresponding to a constant pitch. A
chaotic impulse pattern corresponds to a transient, an initial transient, or
one of tone transition31. Bi-stable states represent multiphonics34 known from
wind instruments like saxophones or clarinets, where a player produces two or
more pitches with the instrument by choosing complex fingerings or playing at
minimum or maximum blowing pressure necessary for normal playing.

The IPF always assumes an arbitrary reference point. Investigating the role of
different parts of a classical guitar in the initial transient of a guitar
tone, for each guitar part, like top plate, back plate, inclosed air or ribs,
a unique IPF using these reference points results in a set of impulses
patterns, one for each guitar part. Relating these IPF with the vibration of
the parts calculated with a Finite-Difference Time Domain (FDTD) method
solving the differential equations for plates and air on a guitar geometry
shows a close correspondence between the IPF and the FDTD transients in terms
of length and complexity8.

The IPF can predict the highly complex behavior of zither initial transients
when varying the table the zither is placed on, a crucial part of zither sound
production. Different strings acting over three zither feet onto different
tables result in very different initial transient lengths. Linear measurement
techniques cannot explain these initial transient lengths in terms of
impedance or eigenmodes. The IPF, on the other hand, is very precisely
modeling the system32.

The minimum bow force needed to arrive at a regular sawtooth motion in violin
or cello bowing has been underestimated with existing models. Again, the IPF
comes very close to experimentally measured minimum bowing pressures much
better than existing models30. This is due to the self-organizing nature of
the bowing process which is not reflected with previous linear models but met
with the IPF.

These reference points have also been proved to correspond with measurements
of the Laotian wind instrument _khaen_24, a set of free reeds attached to
tubes. When playing with low pressure, the instrument sounds with the lowest
eigenfrequency of the reed. From a certain blowing pressure threshold on, the
instrument sounds with the frequency of the tube, which is the normal playing
type. An IPF taking both reference points can understand the role and
interaction of reed and tube and the reason for one of them forcing the other
to go along its eigenfrequencies35.

The IPF has also successfully been implemented in music psychology by modeling
two musicians playing together, where one musician is changing the playing
tempo, and the other musician reacts to this change. With only two reflection
points modeled, the IPF reproduces behavioral data in such cases of tempo
following, by finding that musicians need two beats from the past to adapt to
a tempo change31.

Taking the existing brain models into consideration, the IPF is proposed for
two reasons.

First, as it has already proven to work with musical instruments very
precisely by at the same time computationally very cheap, it has the
capability to model large and extensive systems. This might be a brain; this
might be a society of brains and cultural artifacts like musical instruments
in a musical ensemble, this might also be cultural large-scale entities like
societies or ethnic groups. The scale-free property of the method thereby
allows to focus on tiny details or model large entities in their general
behavior.

Second, it does not differentiate between the kind of impulses but takes sound
impulses in musical instruments and spike trains in the brain as their
physical reality, electric fields, and interactions. This allows reconsidering
the gap between psychology and physics simply by reducing both to electrical
fields. As this holds even for the ecosystem as well as for matter in general,
the IPF might have the capability to model large systems within one dynamical
formulation.

To arrive at such a large-scale model, the IPF must be systematically studied
as a neural model. Therefore, the present paper applies the IPF in parameter
space to examine its behavior and usability as a brain model.

##  II Method

The IPF brain model is presented and tested using a parameter space discussed
below. Next to the inclusion of the relation between inhibitory and excitatory
neuron concentration, four cases are studied, where only the last, the action
case, is realistic. Still, to arrive at an estimation of the realistic
behaviour of the model with and without plasticity and with and without
external input, all cases are equally discussed. Note that the input is
performed in this paper as a separate IPF of a musical instrument. In the long
run, the instrument and the brain are not to be taken as separate entities,
and IPFs, including both are aimed for. Still, for the sake of understanding a
Brain IPF working on its own, in this paper, they are still implemented as
separate entities.

###  A Brain Impulse Pattern formulation (IPF)

The brain is modeled using N = 50 neurons. Each neuron is a reflection point,
returning impulses from a starting neuron, a viewpoint neuron. The system
state of the viewpoint neuron is g, which represents a time period and
amplitude strength. Each reflection neuron i has a damping $\alpha_{i}$. The
IPF, then is

$g_{t}=g_{t-1}-\frac{1}{N}\left|\sum_{i=2}^{N}P_{i}\ln\alpha_{t}^{i}\
g_{t-i}\right|+w_{2}\ g^{I}_{t}\ .$ (1)

Here the viewpoint neuron is i=1. Therefore, the reflections come from neurons
i=2,3,4,…N. Pi are the polarizations of the neuron, where Pi = 1 is an
excitatory and Pi = -1 is an inhibitory neuron. Note that the sum of all
reflections is normalized using the amount of neurons N. The model is discrete
with time steps t=0,1,2,3… Therefore, the earlier states of the viewpoint
neuron, which this neuron has sent out to the other neurons, are returning
after a delay in a damped and polarized form.

Eq. 1 takes the absolute value of the sum. In the case of negative values of
$\alpha_{t}^{i}\ g_{t-i}$, the logarithm would be complex, and therefore g
would become complex. As the cause of the logarithm in the IPF model is
exponential damping, an exponential function with negative, real exponent833,
a complex value of the exponent would also mean a harmonic oscillation, caused
by the imaginary part of g, next to the exponential decay, caused by the real
part of g. In other words, a negative $\alpha_{t}^{i}\ g_{t-i}$ would start a
neural self-oscillation. Now, taking the absolute value of the sum means still
taking the exponential damping, the real part of g, while also including the
frequency of that oscillation, the imaginary part of g. As the system
parameter g is a time interval, including also the imaginary part of g fits
perfectly well into the interpretation of g. Indeed, when using the absolute
value of the sum, negative values of g do not occur, and self-oscillation is
turned off entirely in the present mode.

Taking the absolute value of the sum in Eq. 1 also turns negative real parts
of g into positive ones. Indeed, when wanting to get rid of the imaginary part
of g, one could think of only using the real part of the sum. Still, this
would lead to a negative g. But negative g are unphysical, as they would mean
negative time intervals and negative amplitudes. Still, this is not too much
of a problem, as $Re[\ln(-x)]=Re[\ln(x)]$, and therefore negative g would lead
to the same system behavior. Negative g would also not lead to a positive
exponent and a blow-up of the iteration, as the logarithm turns negative g
into complex numbers, therefore, into oscillations, as discussed above. In any
case, negative g are also not occurring when taking the more plausible
absolute values of the sum as proposed in this paper. Still, future extensions
of the Brain IPF could consider self-oscillations.

###  B Plasticity Model

The plasticity of each neuron is calculated for each time step t. Plasticity
means a change in the damping parameter $\alpha_{i}$, where each time step t
then might have a different damping $\alpha_{t}^{i}$. Note that in the IPF
reasoning, the damping is originally $1/\alpha$, which, for the sake of
convenience, is skipped, now using $\alpha$ instead.

For each time step, the new damping is calculated like

$\alpha_{t}^{i}=\left|\alpha_{t-1}^{i}+w_{1}\ln(1+(g_{t-i}-g_{t}))\right|\ .$
(2)

If the reflection point neuron t-i has the same value $g_{t-i}$ as the
viewpoint neuron value $g_{t}$ the logarithm becomes zero, and no change in
the damping $\alpha_{t^{i}}$ happens. If the reflection point neuron t-i has a
larger value than the viewpoint neuron, the logarithm becomes larger than
zero, and $\alpha_{i}$ increases. Otherwise, the logarithm assures a negative
influence, and $\alpha_{i}$ decreases. The plasticity process is generally
modeled using a constant $w_{1}$. Therefore, plasticity can be switched off in
the model by using $w_{1}=0$. To examine different model behavior, $w_{1}$
will systematically be altered, shown below. Again, the absolute value of
$\alpha$ is used, not allowing negative or complex values. This, again, does
not change the model behavior due to the logarithms used. Still, positive
values are more convenient. Indeed, negative arguments of the logarithm in the
simulations shown below appear very rarely and are additionally suppressed by
using the absolute value.

###  C External Musical Instrument Input IPF

To examine the reaction of the Brain IPF to an external input of a musical
instrument, a simple IPF is used, which models wind instruments or a piano in
the case of normal playing conditions. The time development of the system
parameter GI

$g^{I}_{t}=g^{I}_{t}-\ln\alpha^{I}\ g^{I}_{t-1}\ $ (3)

is used with $g^{I}_{0}=1$ and $\alpha^{I}=1.8$. This leads to a converged
$g^{I}$ in time and a stable tone with constant pitch after a short initial
transient phase of about 12 iterations. This musical instrument is fed into
the Brain IPF as shown in Eq. 1 with a coupling constant $w_{2}$. Again, for
$w_{2}=0$, this input can be switched off.

###  D Parameter Space

To systematically examine the behavior of the Brain IPF and its usability as a
brain model for music, four cases, each with a certain parameter space, are
examined:

  * •

Static: Brain IPF without plasticity and without instrument input

  * •

Plasticity: Brain IPF with plasticity but without instrument input

  * •

Input: Brain IPF without plasticity but with instrument input

  * •

Active: Brain IPF with plasticity and with instrument input

In the plasticity case the plasticity strength $w_{1}$ is varied as $0\leq
w_{1}\leq 2$. In the input case the instrument input strength $w_{2}$ is
varied like $0\leq w_{2}\leq 2$, both in steps of 0.1. In the active case,
both $w_{1}$ and $w_{2}$ are varied accordingly. So the plasticity and input
cases are special cases of the active case with $w_{1}=0$ or $w_{2}=0$
respectively.

The initial values for the reflection strength $\alpha$ are randomly chosen
for all neurons between $0\leq\alpha_{i}\leq 1$ for i = 2,3,4…N, except for
the viewpoint neuron, which has $\alpha_{1}=1$ and $P_{1}$ = 1. The initial
choice of polarization, whether a neuron is excitatory or inhibitory, depends
on the relation $C_{ex/in}$ between both neuron groups. Physiologically,
10-20% of neurons are inhibitory14. Still, locally in certain brain regions,
this relation might show larger deviations. To get an overview and understand
the model behavior over the whole range of possible concentrations, relations
$0\leq C_{ex/in}\leq 1$ are varied, again in steps of 0.1 %.

The IPF might show considerably different behavior for different randomly
chosen values of $\alpha$ and $g_{0}$, the initial value of the system
parameter. To account for this, for each parameter combination of the
parameter space, R=1000 cases were randomly chosen. Thereby, $0.5\leq
g_{0}\leq 2$ was randomly determined. To follow the model’s output when making
changes in the four cases or the parameter spaces, at first, 1000 sets of
$\alpha$ with $g_{0}$ were produced. Then, for all cases and all parameter
combinations in the parameter space, always these 1000 sets were used. This
allows the following model behavior changes independent of the randomly chosen
sets.

###  E Detection of System Behavior

The study aims to discuss the adaptation of the brain to an external stimulus.
Adaptation is defined as the convergence of the model. Convergence is achieved
if

$g_{t}=g_{t-1}\ ,$ (4)

where

$\frac{1}{N}\left|\sum_{i=2}^{N}P_{i}\ln\alpha_{t}^{i}\ g_{t}\right|=w_{2}\
g^{I}_{t}\ ,$ (5)

therefore, the input and the reflections cancel out.

Note that in Eq. 5, the system parameter g is that of $g_{t}$, omitting the
delays at t-I, which means the convergence of g down to N delayed reflections.
The input in the present study is a musical instrument IPF which converges
after about 12 iterations. Therefore $w_{2}\ g^{I}_{t}=const$ thereafter. As
the system parameters, g and $g^{I}_{t}$ are constant at convergence,
condition Eq. 5 means a balance between g and the $\alpha_{t}^{i}$. As the
reflection strength, $\alpha$, is subject to plasticity, a closed analytical
solution of the system is no longer possible. Therefore, an iterative method
is used below to estimate the system behavior.

The output of a Brain IPF is a time series of the system parameter g and a
time-dependent reflection strength $\alpha_{t}^{i}$. The cases without
plasticity have constant $\alpha_{t}^{i}$ throughout. Still, the plasticity
and the active cases also have a time dependency of $\alpha_{t}^{i}$.

To account for all cases discussed above, the system parameter g is written as

$g^{w_{1},w_{2},C_{ex/in},i}_{Case,t}\ ,$ (6)

Where Case is one of the four cases discussed above, $w_{1}$ and $w_{2}$ are
the plasticity and instrument input strength, respectively, $C_{ex/in}$ the
relation of inhibitory vs. excitatory neurons, i is the case of randomly
chosen $\alpha$ and $g_{0}$, and t is the iteration time point. Only the
active case uses both $w_{1}$ and $w_{2}$. With the plasticity and input
cases, only one parameter is shown as an index, and in the static case, both
indexes are omitted. This is easy to read as $C_{in/ex}$ is always given with
a percent sign.

The development of g can basically be of four types which might appear at the
same time or adjacent:

  * •

Converging: a constant value of g is reached over time

  * •

Periodic: a periodicity appears, more or less stable

  * •

Complex: noise-like time series

  * •

Amplitude jumps: sudden amplitude jumps appear, with overall periodic or
complex behavior

Figure 1: Examples of time series g of the Brain IPF showing a) convergence,
b) divergence with periodicity with complexity, c) convergence with complexity
and sudden amplitude jumps, and d) divergence after a time of periodicity.

Fig. 1 shows four examples for time series of the system parameter g for
random $g_{0}$, $\alpha$, and $C_{in/ex}$. In Fig. 1 a) a converged case is
shown. In b) a non-converged case with a periodicity and much complexity is
displayed. The c) plot shows a case of slow convergence with sudden amplitude
jumps, periodicity, and complexity. Finally, in d) a case of periodicity
appearing after a complex initial transient is shown which suddenly diverges.
Clearly, a precise association to one of the phenomena appearing is only
possible with the convergence case which is used in the present paper to
address adaptation.

Of course, these types can be differentiated even more and are not perfectly
defined. Still, this study’s purpose is to adapt the brain to an external
musical instrument input. As we know that this input is periodic after an
initial transient, which as IPF, means a convergent system parameter g, in
this paper, we concentrate on the detection of a converging time series of
brain dynamics. Therefore, we only detect if an IPF is converging or not.

Convergence is detected by calculating the standard deviation of g over the
last 200 values of g. All IPFs in this paper are calculated with 50 reflection
points. As the system parameter g also is a time interval, the length of an
IPF with T iterations depends on the resulting g. So e.g., if a musical tone
with a fundamental frequency of 100 Hz is used as input, the model considers
immediate brain responses within 0.5 seconds. Of course, due to impulses
traveling through all reflection points, the ’echo’ of a system state g or an
external input in the system is much larger. The choice of 50 reflection
points has been determined beforehand by varying this amount from one up to
1000 reflection points. There a convergence of the system behavior up to 50
reflections appeared. Still, a systematic evaluation of the amount has yet to
be performed which might be necessary for future work.

The neocortex frequencies are measured with reasonable strength and associated
with cognitive content only up to about 100 Hz with the highest gamma-band
present from about 30-70 Hz. In the auditory pathway frequencies, up to 4 kHz
are processed from the cochlear on7, but beyond the A1 region of the auditory
cortex, frequencies are much lower. Still, in future studies of the auditory
pathway, the amount of reflection points might need to be increased to cover
higher frequencies.

Convergence of IPFs is calculated as standard deviations over all R cases at
iterations 800 to 1000 like

$S_{Case}^{w_{1},w_{2},C_{in/ex}}=\frac{1}{R}\sum_{j=1}^{R}std(g^{w_{1},w_{2},C_{in/ex},j}_{Case,800-1000})\
,$ (7)

For j=1,2,3,…R, where std is the standard deviation. When averaging over
$w_{1}$ and $w_{2}$ in those cases these parameters are varied we arrive at
$\overline{S}_{Case}^{C_{in/ex}}$.

The mean values for the system parameter themselves are respectively

$M_{Case}^{w_{1},w_{2},C_{in/ex}}=\frac{1}{R}\sum_{j=1}^{R}mean(g^{w_{1},w_{2},C_{in/ex},j}_{Case,800-1000})\
,$ (8)

and $\overline{M}_{Case}^{C_{in/ex}}$.

In the model, 50 reflection points are defined. To arrive at an overview of
the variations of the reflection strength $\alpha$ when adding plasticity to
the model, a mean is defined as

$\overline{\alpha}^{w_{1},w_{2},C_{in/ex}}_{Case}=mean(\alpha_{t=1000,Case}^{i,w_{1},w_{2},C_{in/ex}})\
,$ (9)

where $\alpha_{t=1000,Case}^{i,w_{1},w_{2},C_{in/ex}}$ are the $\alpha$ values
at the latest iteration time point t = 1000 for all cases and plasticity and
input strength variations $w_{1}$ and $w_{2}$ respectively, where present in
the cases. Again, the variable $1\leq i\leq 50$ represents the $\alpha$ at the
respective reflection point in the model, as discussed above. So
$\overline{\alpha}^{w_{1},w_{2},C_{in/ex}}_{Case}$ is integrating over all
reflection points i.

It is also interesting to compare earlier and later reflection point strength.
This is computed for the plasticity case, where overall $w_{1}$ is integrated
like

$\overline{\alpha}^{i,C_{in/ex}}_{Plasticity}=mean(\alpha_{t=1000,Case}^{i,w_{1},w_{2},C_{in/ex}})\
.$ (10)

Another measure is the time at which convergence has been established. This is
again calculated as a mean over R=1000 cases of randomly chosen $\alpha$ and
initial system parameter $g_{0}$ values like

$T_{Case}^{w_{1},w_{2},C_{in/ex}}=\sum_{j=1}^{Rconv}mean\left(P^{w_{1},w_{2},C_{in/ex},j}_{Case}\right)\
,$ (11)

where $P^{w_{1},w_{2},C_{in/ex},j}_{Case}$ are the time points the time series
g has converged. The convergence time point is chosen to be the first time
point where $mean\left(g^{w_{1},w_{2},C_{in/ex},j}_{Case,i\ to\
i+10}\right)<0.01$. So the mean is taken over ten adjacent values of g. The
threshold of 0.01 is chosen concerning the overall range of appearing values
of g. The input sound, also an IPF, reaches this convergence criteria after 12
iterations. The mean for convergence is not taken over all R=1000 cases but
only over those meeting the convergence criteria. All others are considered
not to converge.

Finally, the minima convergence time points are chosen like

$\underline{T}_{Case}^{w_{1},w_{2},C_{in/ex}}=min\left(P_{Case}^{w_{1},w_{2},C_{in/ex},j}\right)\
.$ (12)

##  III Results

For all R=1000 sets of $\alpha^{i}$ and $g_{0}$, all four cases with all
parameter variations $w_{1}$ and $w_{2}$ and all inhibitory/excitatory neuron
concentrations $C_{ex/in}$ IPFs are calculated and convergences
$S_{Case}^{w_{1},w_{2},C_{in/ex}}$, $M_{Case}^{w_{1},w_{2},C_{in/ex}}$,
$\overline{\alpha}^{w_{1},w_{2},C_{in/ex}}_{Case}$,
$\overline{\alpha}^{i,C_{in/ex}}_{Plasticity}$,
$T_{Case}^{w_{1},w_{2},C_{in/ex}}$, and
$\underline{T}_{Case}^{w_{1},w_{2},C_{in/ex}}$ and the means
$\overline{S}_{Case}^{C_{in/ex}}$ and $\overline{M}_{Case}^{C_{in/ex}}$ are
determined. To get an overview, at first, the variations of $C_{in/ex}$ are
discussed concerning overall convergence. Then, system behavior with varying
plasticity and input strengths are discussed followed by a discussion of the
shape of adapted reflection point strength. Finally, convergence time is
discussed in terms of short-time memory effects.

###  A Overall Model Convergence Behavior

A typical relation of inhibitory vs. excitatory neurons in the brain is about
10-20%14. Still, for an overview, it is of interest how the model behaves with
all possible concentrations. Indeed, in some cases, when modeling only a part
of the brain, different relations might be present.

Fig. 2 shows at the top plot $\overline{S}_{Case}^{w_{1},w_{2},C_{in/ex}}$,
the averaged standard deviations of the system parameter g over iterations
800-1000. This parameter shows the overall convergence behavior of the model.
As a first rough overview, the averaged convergence is taken over all R=1000
cases of $\alpha$ and $g_{0}$ and over the variations of $w_{1}$ and $w_{2}$
when present in the respective cases.

In all cases, a peak at a 50% relation of $C_{in/ex}$ is found. The static
case has the strongest convergence, nearly the same when giving an input to
the model. This is expected as the input itself has fast convergence. When
adding plasticity to the model, the convergence is decreased tremendously. The
brain model’s activity is counteracting the static case’s convergence
behavior. The convergence is again much stronger when adding the input, but
without arriving at the original convergence strength of the static case.

Figure 2: Standard deviation $\overline{S}_{Case}^{C_{in/ex}}$ (top) and mean
$\overline{M}_{Case}^{C_{in/ex}}$ (bottom) of the system parameter g for
iterations 800-1000, averaged over all R=1000 cases of randomly chosen
$\alpha$ and $g_{0}$ and variations of coupling strength $w_{1}$ and $w_{2}$
for the static, plasticity, input, and active cases. Therefore the plots give
a rough overview of the convergence of the model. When varying the amount of
inhibitory vs. excitatory neurons in the model, for convergence, a peak at
50%, and for mean, a minimum appears in all cases. Still, for the
physiologically realistic active case, this peak is the smallest and overall
smooth behavior is found.

An opposite behavior is found for the mean system parameter
$\overline{M}_{Case}^{w_{1},w_{2},C_{in/ex}}$ shown in the bottom plot of Fig.
2. There, a minimum is found in all cases at $C_{in/ex}$ = 50 %. This points
to a balance of inhibitory and excitatory neurons at 50%, leading to low
values of the system parameter.

Again, the plasticity case is the most extreme, but the values are lowest
here. The input case has the largest values as expected, as the input gives
additional energy into the system. The active case is again medium and,
therefore, again balancing between the plasticity and the input case.

Therefore, the model behaves as expected from a neural network in general.

###  B Convergence when varying Plasticity
Strength

In the plasticity case, only the plasticity strength $w_{1}$ is varied. Fig. 4
shows the convergence $S_{plasticity}^{w_{1},w_{2},C_{in/ex}}$ for 0% $\leq
C_{in/ex}\leq$ 50% variations of the relation between inhibitory and
excitatory neuron concentration. For realistic concentrations of 10 - 20%, a
strong decrease of convergence is found with low plasticity strength $w_{1}$.
Then, with $w_{1}>0.3$, convergence is strong again. With higher
concentrations, starting from 30%, this peak is much eased and appears with
higher $w_{1}$. The mean system parameter $M_{Plasticity}^{w_{1},C_{in/ex}}$
is decreasing with increasing convergence.

This behavior points to an instability caused by the interplay of differently
delayed neurons.

Figure 3: Model convergence when varying plasticity strength $w_{1}$ in the
plasticity case (top plot) and the mean system variable g in this case (bottom
plot). The convergence in the plasticity case is less converging for weak
plasticity strength $w_{1}$ and realistic $C_{in/ex}$ concentrations.
Accordingly, $M_{Plasticity}^{w_{1},C_{in/ex}}$, the mean of the system
variable g is decreasing with increasing convergence.

###  C Convergence when varying Input
Strength

In the input case, the situation is more apparent. Increasing $w_{2}$ leads to
stronger convergence. As found before for the input case, $C_{in/ex}$
concentrations around 50% converge less but are also eased with higher
$w_{2}$. Still, the mean system parameter $M_{Input}^{w_{2},C_{in/ex}}$ is
increasing with decreasing convergence. This is the opposite behavior compared
to the plasticity case, where the system parameter decreased with increasing
convergence. The additional energy of the input to the system can explain
this.

###  D Neuron Reflection Point Strength
Delay-Dependency

When plasticity is added to the model, in the plasticity and the active case,
the convergence of $\alpha$ is of interest. At t = 1000 iterations, so at the
end of the chosen simulation length, the alpha randomly selected at the
beginning will have converged in all converging cases of the system parameter
g.

Figure 4: Model convergence when varying the input strength $w_{2}$ in the
input case. The input case behaves as expected by increasing convergence with
higher $w_{2}$. Contrary, $M_{Input}^{w_{2},C_{in/ex}}$ is increasing with
higher input strength $w_{2}$.

Fig. 5 shows $\overline{\alpha}^{i,C_{in/ex}}_{Plasticity}$, the integration
over all plasticity strength $w_{1}$. As $C_{in/ex}$ determines the amount of
positive and negative reflection points, the left side of the plot is positive
while the right side is negative, as expected. Still, the reflection point
strength is strongest around reflection point 30 in all cases. The uneven
contour lines point to a large variation of this behavior. Still, overall in
this model, reflections around reflection point 30 has the largest impact on
the system behavior when the system adapts to itself without any input.

Figure 5: Mean of $\alpha$ at the 50 reflection points at the latest time
point t = 1000 for the plasticity case and all $C_{in/ex}$. For a greater
amount of excitatory neurons on the plot’s left side, the values are positive
and vice versa. A peak of reflection strength is around reflection point 30 in
all cases.

###  E Neuron Reflection Point Strength with changing Plasticity and Input
Strength

The adaptation of the reflection point strength
$\overline{\alpha}^{w_{1},w_{2},C_{in/ex}}_{Active}$ when varying both
plasticity and input strength $w_{1}$ and $w_{2}$ respectively is shown in
Fig. 6 for four inhibitory vs. excitatory neuron concentrations $C_{in/ex}$.
These are integrated over all reflection points.

The range of values decreases with increasing $C_{in/ex}$ from over 30 down to
about 3.5. The large values in the 0% plot on the top left are mainly found
with large $w_{1}$ and $w_{2}$, while for small plasticity and input strength
values around 3-9 are found. These values decrease with increasing
concentration. Also, the peak of $w_{1}$ with low $w_{2}$ is changing from
around 0.2 at 0% to about 1.1 at 30%. This peaking behavior is also found with
$w_{2}$, although not that strong. In the 30% case and constant $w_{1}$,
$\overline{\alpha}^{w_{1},w_{2},30\%}_{Active}$ in- and decreases with
increasing input strength $w_{2}$.

This is pointing to a more realistic model behavior with realistic $C_{in/ex}$
of 20 - 30%.

Figure 6: Mean neuron reflection point strength when varying plasticity and
input strength $w_{1}$ and $w_{2}$ respectively for different inhibitory vs.
excitatory neuron concentrations $C_{in/ex}$.

###  F Convergence Time

The convergence time has been defined as a mean and a minimum time. It is
discussed here for the active case, as the input and plasticity cases are only
special cases of the active case with $w_{1}=0$ and $w_{2}=0$, respectively,
and therefore included in the discussion. The overall behavior of the mean
convergence time is similar to that of the plasticity
$\overline{\alpha}^{w_{1},w_{2},C_{in/ex}}_{Active}$ shown in Fig. 6 and
therefore omitted here. The minimum convergence times
$\underline{T}_{Active}^{w_{1},w_{2},C_{in/ex}}$ are shown in Fig. 7. Note
that the mean convergence times are calculated from only those R=1000 cases
where between iterations 800-1000 convergence has appeared. All other cases
are considered non-converging and excluded here. Therefore, discussing a
maximum convergence time is not reasonable.

The mean convergence time $T_{Active}^{w_{1},w_{2},C_{in/ex}}$ covers a range
from about 100 to 500 iterations. Taking 100 Hz as the fundamental frequency
of the input sound and an the iteration time interval of 10 ms per iteration
of the system parameter g, 500 iterations corresponds to five seconds. This
corresponds to short-time memory time. Although much smaller convergence times
exist, the model is reasonably within typical brain behavior. Therefore, the
Brain IPF explains short-time memory length using the physiological threshold
of strong neural activity only up to about 100 Hz, as found in EEG signals.

Taking a look at the minimum time point
$\underline{T}_{Active}^{w_{1},w_{2},C_{in/ex}}$ as shown in Fig. 7, for
$C_{in/ex}$ = 20 % and 30 %, the distribution is different from the mean
cases, where a convergence of about 100 iterations, so about 1 second is found
nearly all over variations of $w_{1}$ and $w_{2}$. Exceptions are only found
for $w_{1}=0$. This is different for the 0 % and 50 % cases. For 0 %, small
$w_{1}$ lead to an increase of the minimum time, while in the 50 % case, an
increase of the input strength $w_{2}$ leads to a decrease of this minimum
time. So in the realistic cases of 20 % and 30 % cases of inhibitory vs.
excitatory neuron concentration, the minimum convergence possible is basically
independent of plasticity and input strength.

Figure 7: Minimum convergence time
$\underline{T}_{Active}^{w_{1},w_{2},C_{in/ex}}$ of the active case when
varying $w_{1}$ and $w_{2}$ in number of iterations. Taking a fundamental
frequency of the input sound of 100 Hz, 100 iterations correspond to 1 second.
Therefore, convergence times fit within short-time memory time.

##  IV Conclusion

Viewing the brain as a system of spike bursts sent out by an arbitrary
reference point to multiple other brain regions, which return the bursts in a
damped manner, as implemented in the IPF Brain model discussed in this paper,
leads to reasonable behavior. Converged states with a periodicity are
possible, next to complex and chaotic states. With no sensory input, the case
with plasticity tremendously increases the amount of complex brain activity
while reducing the overall system energy compared to the case without
plasticity. With sensory input added, the system strongly adapts to this input
while simultaneously increasing overall system energy. Still, sensory input
with plasticity means more complexity than sensory input without plasticity.
Therefore, plasticity leads to less adaptation and more behavior intrinsic to
the brain. This corresponds to an active, creative adaptation of the brain to
an external stimulus rather than a passive _tabula rasa_.

The amount of inhibitory vs. excitatory neurons in the brain is between 10 -
20%. Still, in single brain units, the amount might be different. Here again,
the active case of plasticity adapting to an external sensory input differs
from all other cases. While the amount of complexity shows a maximum at
$C_{in/ex}$ = 50%, the active case has a minimum of complexity for this region
of 10 - 20%. The Brain IPF, therefore, finds this physiological finding in a
minimization of brain complexity when adapting to sensory input.

This preference for $C_{in/ex}$ = 10 - 20% also appears in the plasticity case
when varying the plasticity strength $w_{1}$. For 0% $\leq C_{in/ex}\leq$ 20%
complexity decreases strongly for $w_{1}<~{}0.3$, although for very low
$w_{1}$ in this region complexity is much stronger compared to higher
$C_{in/ex}$. As plasticity strength might vary for different brain regions,
this behavior means a effective variability of brain behavior by varying
$w_{1}$, which might also appear temporally, a variation beyond the scope of
this paper.

The brain regions’ reflection strengths $\alpha$ are systematically
distributed after convergence with a maximum strength at about reflection
point 30, increasing to this maximum from the reference point on and
decreasing towards reflection point 50. This is strongest for low $C_{in/ex}$.
When using a sensory input with a periodicity of 100 Hz, a maximum of 30
reflection points corresponds to a maximum reflection strength around 300 ms.
This is a timescale known from Event-Related Potentials (ERPs), where a
reaction of a stimulus is found to be present in time intervals of roughly
100-400 ms, as e.g., found with Mismatch Negativity (MMN) experiments of
musical syntax found in the Broca region27, binaural Interaural Cross-
Correlation (IACC) spatial perception neural mastoid response36, or used in
understanding music perception of cochlear implant patients41.

Considering the convergence time points, the model meets the short-time memory
time-span of up to about five seconds, often associated with an echoic
memory42. A minimum with a ,reasonable input and plasticity strength of about
one second is found. All this appear when taking 100 Hz as the fundamental
frequency of the input sound and therefore meet the physiological finding of
strong periodicities in the cortex only up to about 100 Hz. Thus, the model
finds short-time memory length to come into place due to this 100 Hz maximum
and the basic idea of the brain acting as an interchange of damped and delayed
reflected impulses.

Other time scales are known and important in the brain and in auditory
perception. A periodicity of 50 Hz of oscillating neurons was found to be the
basis of perception and cognition14402. This frequency is associated with the
gamma-band which also convers higher frequencies up to about 100 Hz. Also, 50
Hz is half the frequency used in this model. Neurons firing at 50 Hz on their
own and in asynchrony with other neurons also oscillating at 50 Hz will lead
to a period-doubling of the overall oscillation leading again to 100 Hz.
Therefore, the findings presented in this paper support a periodicity of 50 Hz
more than arguing against it.

The model is that of a neocortex, as it is referenced to EEG signals taken
from the neocortex. Of course, adapting to an external sound input while
perceiving a musical pitch is much faster and within the range of
milliseconds. The second integration time of auditory and visual perception is
around 50 ms, where successive events with shorter intervals are no longer
perceived as separate. In terms of auditory perception, this leads to the
lowest frequency of about 20 Hz, where rhythm perception ends, and pitch
perception starts. Although pitch perception is not understood in terms of a
general agreement until now, many findings point to a subcortical perception,
including spaciousness or timbre perception2. Discussing this is beyond the
scope of this paper. So a Brain IPF of the auditory pathway would need to use
a much shorter time interval compared to the 100 Hz used in the present model
of the neocortex.

The Brain IPF model, therefore, behaves reasonable within existing findings of
brain activity. Its advantage compared to existing models is its very low
computational cost and flexibility in terms of variations of plasticity
strength, sensory input, or time scales. Another major advantage is a
formulation already found working with musical instruments, so with cultural
artifacts. This opens the possibility to model ensembles of brains and musical
instruments or in the future even larger entities like societies or ecosystems
within one model, which is computationally fast, highly flexible, and very
precise in terms of system predictions, as already found with musical
instruments.

##  V Acknowledgement

We thank Robert Mores and Simon Linke for their valuable comments on the
paper.

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