Statistical Crossover and Nonextensive Role on the Neuronal Short Term Depression

Abstract

The theoretical basis for the neuronal coding, associated with the short term degradation in synaptic transmission, is matter of debate in the literature. In fact, electrophysiological signals are characterized as inversely proportional to stimulus intensity. Among theoretical descriptions for this phenomenon, models based on 1/f-dependency are employed to investigate the biophysical properties of the short term synaptic depression. Thus, considering 1/f-model, as starting point for describing synaptic depletion, we adopt a paradigmatic \textit{q}-differential equation obtain a generalized formalism for investigation of nonextensivity in this specific type of synaptic plasticity. Our analysis reveal nonextensivity in data from electrophysiological recordings, also uncovering a statistical crossover in neurotransmission, which gives additional support to the hypothesis of heterogenous release probability of neurotransmitters. On the other hand, the 1/f-model achieved a satisfatory agreement with data only at lower frequency stimulations. In resume, the present work presents a method to demonstrate that short term depression is ruled by a nonextensive behavior. Our findings also better conciliate morphological and electrophysiological investigations in a coherent biophysical scenario.

Full text

1 23
Journal of Biological Physics
ISSN 0092-0606
Volume 44
Number 1
J Biol Phys (2018) 44:37-50
DOI 10.1007/s10867-017-9474-3
Statistical crossover and nonextensive
behavior of neuronal short-term depression
A.J.da Silva, S.Floquet &
D.O.C.Santos

1 23
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J Biol Phys (2018) 44:37–50
https://doi.org/10.1007/s10867-017-9474-3
ORIGINAL PAPER
Statistical crossover and nonextensive behavior
of neuronal short-term depression
A. J. da Silva1·S. Floquet2·D. O. C. Santos1
Received: 10 February 2017 / Accepted: 20 September 2017 / Published online: 13 October 2017
© Springer Science+Business Media B.V. 2017
Abstract The theoretical basis of neuronal coding, associated with short-term degradation
in synaptic transmission, is a matter of debate in the literature. In fact, electrophysiolog-
ical signals are commonly characterized as inversely proportional to stimulus intensity.
Among theoretical descriptions of this phenomenon, models based on 1/f -dependency are
employed to investigate the biophysical properties of short-term synaptic depression. In
this work, we formulate a model based on a paradigmatic q-differential equation to obtain
a generalized formalism useful for investigation of nonextensivity in this specific type of
synaptic plasticity. Our analysis reveals nonextensivity in data from electrophysiological
recordings and also a statistical crossover in neurotransmission. In particular, statistical
transitions provide additional support to the hypothesis of heterogeneous release proba-
bility of neurotransmitters. On the other hand, the simple vesicle model agrees with data
only at low-frequency stimulations. Thus, the present work presents a method to demon-
strate that short-term depression is not only governed by random mechanisms but also by
nonextensive behavior. Our findings also conciliate morphological and electrophysiological
investigations into a coherent biophysical scenario.
Keywords Nonextensivity ·Crossover statistics ·Synaptic depression ·Neural plasticity
The original version of this article was revised: The authors apologize for the following errors
published in the article. However, these errors do not modify the main assumptions in their
work nor affects the discussion (interpretation) of the results. Brasil was changed to Brazil in the
affiliations and some text under the Discussion and Conclusions’ sections were corrected.
A. J. da Silva
adjesbr@ufsb.edu.br; adjesbr@gmail.com
1Centro de Formac¸˜
ao em Ciˆ
encias e Tecnologias Agroflorestais,
Universidade Federal do Sul da Bahia, Itabuna, Bahia. CEP 45613-204, Brazil
2Colegiado de Engenharia Civil, Universidade Federal do Vale do S˜
ao Francisco, Juazeiro, Bahia.
CEP 48902-300, Brazil
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38 A. J. da Silva et al.
1 Introduction
Neural communication is an intricate molecular process that is still not well understood.
Information processing in the central nervous system (CNS) is mainly achieved by spe-
cialized structures called chemical synapses. Synaptic transmission is mediated by one or
more neurotransmitter substances, accomplished in the following steps [1–3]: (1) action
potential triggers opening of voltage-gated calcium channels in the nerve ending; (2) open-
ing of these channels allows an influx of calcium ions into the neuron terminal; (3) on
the active zone (AZ) of the cell membrane, calcium ions trigger vesicle fusion and neu-
rotransmitter release into the synaptic cleft; (4) secreted neurotransmitters diffuse into the
synaptic cleft, reaching receptors located in the postsynaptic neuron. Postsynaptic excita-
tory or inhibitory current (IPSC)orpotential(VPSP) are prompted by neurotransmitters
bound to the postsynaptic receptors. These electrical events are readily assessed by electro-
physiological measurements. However, sustained presynaptic activity does not necessarily
release the same amount of neurotransmitter into the synaptic cleft. Within the synaptic
terminal, vesicles share a crowded environment, forming readily releasable, recycling and
reserve pools. These pools are successively recruited under sustained presynaptic stimula-
tion, which initially promotes fusion of readily releasable or docked vesicles on the AZ.
Higher frequencies promote a release probability increment of recycling and reserve pools,
respectively. In other words, there is recruitment of vesicles from both pools toward synaptic
fusion.
The ability of neurons to change their vesicular dynamics, affecting synaptic strength,
defines neuronal plasticity. For instance, a particular form of neuroplasticity, known as
short-term depression (STD), exhibited by different synapses in the brain, is characterized
by IPSC or VPSP amplitude decrement. This promotes a degradation of synaptic trans-
mission temporal fidelity, controlling the statistical properties of neurotransmission [4]. To
explain this mechanism, models for STD characterization based on 1/f -dependency were
developed to be tested over different experimental paradigms [5]. Nevertheless, such mod-
els show limitations in accurately explaining experimental data. For example, studies show
that neurotransmitter release does not behave like a haphazard process. Therefore, the need
to develop more robust theoretical strategies is evident. In this context, a possible nonex-
tensivity in STD certainly contributes to clarify complex mechanisms involved in synaptic
transmission [6–8]. Bernard Katz and colleagues introduced a statistical pillar for neuro-
transmission, quantifying the vesicular fusion as a Gaussian phenomenon [9]. Additional
reports expanded this statistical description after considering other distribution functions
[10]. Relative to STD, in spite of its limitations, binomial statistics is the conceptual basis to
characterize the degradation of plasticity. However, the exploration of statistical heterogene-
ity in neuroplasticity has not been contemplated using Tsallis Statistics. We hypothesize
that the possible existence of statistical transitions and nonextensivity can overcome the
restrictions of previous models by providing a more general scenario.
Nonextensive statistical mechanics (NSM) describes systems in which the entropy is not
proportional to the system size, a property frequently observed in complex systems that dis-
play long-range interactions or that are out of equilibrium [11]. In this framework, Tsallis
Statistics is successfully employed to investigate a variety of phenomena due to its ability
to model power law phenomena [12]. Although its application is widespread, NSM remains
scarcely applied in studies of the physiology of neurotransmission, despite the confirma-
tion of nonextensivity associated to spontaneous release at the mammalian neuromuscular
junction [13,14]. However, our previous reports included neither brain synapses nor the pos-
sible role of electrical stimulation on nonextensivity. To overcome these limitations, using
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Statistical crossover and nonextensive behavior of the neuronal... 39
electrophysiological results collected at different synapses, we investigate whether there are
nonextensivity and statistical transitions governing the neuronal communication involved in
STD mechanism.
2 Methods
2.1 Experimental data
Before introducing the theoretical analysis, we justify the use of the selected experimental
data (see references for detailed experimental procedures). Electrophysiology, represented
by a family of empirical tools, is largely employed to investigate neuronal activity, used
in clinical examinations and in high-throughput screenings. Electrophysiological record-
ings are regularly applied for in vitro studies, where patch-clamp and extracellular field
potential techniques are employed to understand the substrate of neuronal plasticity.
Among the preparations used in STD studies, one can highlight the auditory and lim-
bic systems. Located in the auditory brainstem, the synapse formed by the calyx of Held
and the main neuron medial nucleus of the mammalian trapezoid body (MNTB) is an
important preparation to study STD due to their large cell size that facilitates empiri-
cal manipulations. Additionally, using the same morphological argument, the avian bulb
of Held and the nucleus laminaris make these models suitable for electrophysiological
recordings, including STD assessment [15,16]. Moreover, the avian endbulb of Held
and the nucleus laminaris are relevant to address important questions in evolutionary
neuroscience and comparative physiology of synaptic plasticity. Thus, these preparations
expand our findings beyond those computed in mammalian species [17]. The hippocam-
pus, part of the limbic system, is a crucial brain area responsible for spatial memory,
learning, and navigation. An empirical advantage is the straightforward process of tis-
sue extraction, which allows cytoarchitecture preservation and accurate visualization of
its different areas, being a highly used preparation for neuronal electrophysiology [18,
19]. Adoption of the same formulation among different synapses is also justified, despite
the functional and morphological particularities involved, due to common features in
exocytosis machinery dynamics and similar electrophysiological response in STD curve.
This strategy also allows investigating how ubiquitous nonextensivity is in the nervous
system.
Summarizing, the data used here were collected from intracellular IPSC and extracellular
VPSP studies carried out in auditory synapses and in the dentate gyrus of hippocampal slices
[20–23]. Importantly, in spite of the methodological differences between extracellular and
intracellular measurements, there is evidence that intracellular electrophysiological proper-
ties can be predicted by extracellular recordings [24]. This argument supports applications
of nonextensive analysis of intra or extracellular electrophysiological recordings.
2.2 Theoretical modeling
2.2.1 Crossover statistics
Boltzmann–Gibbs statistics (BG) states that the entropy additivity law, only valid for
extensive systems, is governed by S=−kP(x)ln P(x). In this case, the exponential
probability density P(x) ∝exp(−x), represents the entropy distribution of noninteracting
systems. Because it considers long-range correlations, NSM brings a generalization for the
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40 A. J. da Silva et al.
classical description, since in their foundations a nonextensive or nonadditive entropy rule
is assumed for Sq, written as:
Sq(A +B) =Sq(A) +Sq(B) +(1−q)Sq(A)Sq(B). (1)
In this case Aand Bare two independent systems with:
P(x,x)A+B=P(x)
AP(x)B.(2)
In this sense, P(x) represents the probability density distribution of the macroscopic
variable x. Therefore, the so-called entropic index qexpresses the magnitude of a nonex-
tensivity operating in the system. The maximization of the q-entropy leads to:
Sq=k
(q −1)1−[P(x)]qdx,q∈R.(3)
Its optimization produces a q-exponential distribution:
Pq(x) ∝ex
q≡[1+(1−q)x]1/(1−q) (4)
if 1 +(1−q)x≥0andex
q=0 otherwise. In the limit q→1 the usual BG entropy is
recovered, S1≡SBG,and(4) converges to the usual exponential distribution, P(x) ∝ex.
(4) can also be obtained from:
dP
dx =−λqPq,λ
q≥0;q≥1(5)
where P=Pq(x) has a solution Pq(x ) =1+(1−q)λqx1/(1−q).Thispowerlaw,
exactly the same q-exponential function showed in (4), was applied to biological systems,
discriminating supperdiffusive patterns in dissociated cells from Hydra and giving a novel
description of internucleotide interval distribution [25,26].
We now discuss the conditions under which varying a macroscopic variable leads to the
changing of a statistical regime governing the form of a probability density distribution.
First, it is necessary to generalize (5) to one that unifies BG statistics and nonextensive
statistics. This is accomplished by the paradigmatic equation [27]:
dP
dx =−μrPr−(λq−μr)P q,r≤q, q ≥1.(6)
For r=q=1,∀λqwe recover BG statistics, that is dP
dx =−λqP,q=1. For μr=0,∀r
or if r=q,∀μrwe recover usual nonextensive statistics, given by (5). In this equation, r
measures the degree of both nonextensivity (r>1) and extensivity (r=1) in the same
sense as the q-index. Solutions of (6) permit to observe a crossover from different statis-
tical regimes from low xvalues, dominated by a q-exponential behavior, to high xvalues
described by r-exponentials [12,28]. Equation (6) was successfully employed to detect
nonextensivity and to determine the statistical crossover in studies of the flux of cosmic
rays [29] and protein folding [28]. Interestingly, under assumptions, this equation can also
be seen as a generalization of Planck statistics [30].
Parameters λqand μrdetermine the values of xmarking the change of a statistical
regime. We observe this with a particular solution for r=1andq>1, which presents
transition from nonextensivity to extensivity [12,28]:
P(x) =⎛
⎜
⎜
⎝
1
1−λq
μr
+λq
μr
exp [(q −1)μrx]
⎞
⎟
⎟
⎠
1/(q−1)
,x≥0.(7)
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Statistical crossover and nonextensive behavior of the neuronal... 41
From this expression, the values of xmarking a given crossover are stated by the following
expressions, for r= q,μr=1λq. In this case, we have a first crossover in x∗
qand a
second crossover in x∗
r=1:
x∗
q=1
λq(q −1),(8)
x∗
r=1=1
μr=1(q −1).(9)
For 1 <r <q, expressions for xin a crossover can be found in [28,29].
2.2.2 Crossover statistics in short-term depression
The amplitude response of the electric signal involved in STD is characterized as inversely
proportional to stimulus frequency. As previously discussed, many theoretical descriptions
of this phenomenon were based on 1/f -dependency, employed to investigate the biophys-
ical properties of STD. Injection of a repetitive stimulation in the pre-synaptic terminal
decreases IPSC or VPSP responses yielding STD. Let us consider a variable Rrepresenting
IPSC or VPSP responses. A formulation based on the 1/f -behavior, used by other authors
to investigate STD data [5], is denominated by the simple vesicle depletion model given by
the following expression:
R=1
1+fpτ (10)
where τis a relaxation time constant toward a steady state, fis the stimulation frequency
and pis the release probability. It is important to mention that this model neglects vesicle
interactions into the synaptic terminal, being consistent with a binomial statistical descrip-
tion. We can further generalize (10) giving it a power law format, introducing an exponent
n:
R=1
1+pτf n
.(11)
If we make n=1
q−1and τp =λ(q −1),whereqis the nonextensive index, a
nonextensive simple vesicle depletion model is obtained:
R=1
1+λ(q −1)f 1
q−1
.(12)
Here, we also make the hypothesis that a q-exponential function governs IPSC or VPSP
responses. As shown above, (12) is a solution of the following:
dR
df =−λRq.(13)
A further generalization of (13) is possible if we introduce another nonextensive index,
rand assume that nonextensivity and statistical crossover occur in STD. This equation is
similar to (6):
dR
df =−μRr−(λ −μ)Rq(14)
With r≤qand q≥1, (14) admits an analytical solution as a family of hypergeometric
functions, which give an approximated solution as they have to be truncated [28]. For this
reason, we choose to numerically integrate (14) to investigate nonextensivity in STD. This
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42 A. J. da Silva et al.
equation also predicts that increasing frequency causes the changing of a statistical regime
governing STD. For example, crossover frequencies, for r= qand r=1 are given by:
f∗
q=1
λ(q −1),(15)
f∗
r=1=1
μ(q −1).(16)
To uncover how parameters λand μchange the shape of Rwe performed simulations
varying those parameters in (14). The results are presented in Fig. 1. Our simulations suggest
that λadjusts the curve concavity or drives the rate of decay in the early phase (lower
frequencies). In contrast, μcontrols the degree of depression relative to the maximum IPSC
or VPSP. This also suggests that μand pcould be related.
Given the information in Fig. 1, it is necessary to write λand μas functions of parame-
ters related to STD. To achieve this, and influenced by previous models, we start listing the
main premises of our model: (1) the synaptic terminal is constantly supplied from an unlim-
ited vesicle reservoir, where vesicles are recruited independently from each other, with rate
κ; (2) vesicles, placed into the readily releasable pool at an active zone, are allowed to inter-
act with each other by the proteins of the exocytotic machinery; (3) vesicles possess a mean
residence time or a relaxation time given by τ; (4) we do not take into account a refilling
rate, that is, we are neglecting endocytosis contribution; (5) the release probability pand
quantal size Qof vesicles are parameters independent of each other. Quantal size is the post-
synaptic response originated from the release of a single vesicle; (6) there is a relationship
between the pand the q-index; (7) we do not consider calcium contribution.
Finally, we suggest a non-extensive physiological mechanism depicted in Fig. 2. Accord-
ing to this illustration, the exocytotic machinery comprises two types of mechanisms.
Beyond the random mechanism, coherent with an extensive scenario, three situations
emerge, as a consequence of non-extensivity: lateral inhibition, facilitated release and total
inhibition.
We propose that λand μare functions of the new parameters τ, relaxation time; κ,
recruitment rate of vesicles; p, release probability; Q, quantal size:
μ=pQκ, (17)
λ−μ=pQ/τ. (18)
This implies μ>0 as this parameter is a product of positive quantities. Given these assump-
tions, we propose an equation for R, incorporating crossover statistics, which resembles one
in the work by Niven [31]. The result is the q-differential equation:
dR
df =−pQκRr−pQ
τRq.(19)
In this case, r≤qand q≥1. Parameters rand qare crossover exponents as presented in
the former section. It is important to stress that (14)and(19) are equivalent.
2.3 Data analysis and optimization
We used WebPlotDigitizer [32] to extract data from articles. Parameters from (14)were
estimated using genetic algorithms (GA), a class of optimization or parameter search algo-
rithms incorporating biological evolution mechanisms [33]. We used GA to find a vector
in parameter space that minimizes the root mean squared difference between experimental
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Statistical crossover and nonextensive behavior of the neuronal... 43
0 50 100 150 200
0
0.2
0.4
0.6
0.8
1
VPSP (normalized)
= 0.001
= 0.01
= 0.05
= 0.1
= 1.0
= 5.0
0 50 100 150 200
0
0.2
0.4
0.6
0.8
1
VPSP (normalized)
= 0.0009
= 0.005
= 0.009
= 0.02
= 0.05
= 0.1
50 100 150 200 250 300
0
0.2
0.4
0.6
0.8
1
VPSP (normalized)
= 0.010
= 0.050
= 0.205
= 0.5
= 0.8
= 2.0
50 100 150 200 250 300
0
0.2
0.4
0.6
0.8
1
VPSP (normalized)
= 0.0005
= 0.0020
= 0.0045
= 0.01
= 0.02
= 0.05
246810
Frequency (Hz)
0
0.2
0.4
0.6
0.8
1
IPSC (normalized)
= 0.30
= 1.00
= 3.99
= 8.0
= 12.0
= 30.0
246810
Frequency (Hz)
0
0.2
0.4
0.6
0.8
1
IPSC (normalized)
= 0.0216
Fig. 1 Left: adjustments, with (14), for several values for λ, keeping the other parameters fixed. Right: same
as left but for several values of μ. Data points correspond to data from: [23] (top), [21] (middle) and [5]
(bottom). According to these simulations, λdrives the rate of decay in the early phase (lower frequencies),
while μcontrols the degree of depression relative to the maximum IPSC or VPSP
data and simulated points from (14). Computer simulations were performed in R-cran and
MATLAB. Numerical calculations were run independently, in the sense that the researchers
interacted with each other only after the parameters were obtained. Even with this strategy,
our results were similar by themselves, which showed the robustness and reliability of our
data analysis.
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44 A. J. da Silva et al.
Fig. 2 Illustration of a STD scenario presenting nonextensive and extensive statistics. Vesicles are recruited
from a reserve pool with a constant rate recruitment (k). On the AZ, interaction absence (V1and V2)orpres-
ence (V3and V4) are possible. The latter may emerge as a consequence of protein sharing from the SNARE
(Soluble NSF Attachment protein REceptor) complex (represented by a black stripe), yielding three possi-
bilities: facilitated release, lateral or partial inhibition or total inhibition. This is the substrate of nonextensive
mechanisms. On the other hand, an independent release is connected to an extensive process
3Results
We first applied (14) to determine either qand rindexes and parameters λand μ. We cal-
culated the release probability, pand κusing (17)and(18) for fixed values of Qand τ
from the literature. Crossover frequencies were calculated using (15)and(16). In experi-
mental protocols of STD, the calyx of Held synapse is characterized by a rapid IPSC decay
followed by a pronounced steady-state region at higher frequencies. The adjustment of data
from the calyx of Held (Fig. 3, top from [20]) provides as the adjusted values q=r= 5.192.
This indicates a purely nonextensive regime, better described by (12), without a statistical
crossover.
We also hypothesized whether nonextensivity could be verified in non-mammalian
synapses by investigating VPSP data from the avian auditory system (Fig. 3, Bottom) [21].
Transitions from nonextensivity (q= 4.326) to extensivity (r= 1.000) are observed with
the first and second crossover frequencies given by f∗
q= 1.467 Hz and f∗
r=1= 75.165
Hz, respectively. Next, we investigated data from the hippocampus [23](Fig.4), which
resulted in estimated parameters q= 7.933 and r= 1.013 with f∗
q= 0.182 Hz and f∗
r=1
= 16.026 Hz. Here we used (8)and(9) to calculate approximate values for the crossover
frequencies.
Our results are summarized in Table 1. Adjustments to the 1/f -based equation, (10),
were only partially achieved in all cases. We conclude: (a) there are statistical transitions in
STD phenomena; (b) nonextensivity is present in auditory system synapses of mammalian
and non-mammalian species; (c) nonextensivity is observed in data from intracellular and
extracellular environments; (d) nonextensivity in dentate gyrus plasticity suggests a rela-
tion between nonextensivity and the neuronal substrate involved in learning and memory of
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Statistical crossover and nonextensive behavior of the neuronal... 45
151015
Frequency (Hz)
0.1
0.5
1
IPSC (normalized)
15 10
Frequency (Hz)
0
0.5
1
IPSC (normalized)
10 110 210 3
Frequency (Hz)
0.1
0.5
1
VPSP (normalized)
100 200 400
Frequency (Hz)
0
0.5
1
VPSP (normalized)
Fig. 3 Auditory system data recorded with the patch clamp technique (log-log and linear-linear scales) and
respective adjustments for both models. Full lines represent fitting with (14), while dashed lines correspond
to adjustments with (10). Top : data points adapted from excitatory IPSC recordings measured by von Gers-
dorff et al. ([20], Fig. 2a), also analyzed in Weis et al. ([5], Fig. 2b) and Trommershauser et al. ([35], Fig.
4a). Bottom: fits using excitatory VPSP data ([21], Fig. 1d). The insets show the linear-linear representation
of the data. In the bottom inset,thevertical line marks the crossover frequency at 75.165 Hz. The first
crossover at 1.467 Hz is not visible
other hippocampal areas; (e) diversified statistical transitions point out that although neu-
rotransmitter secretion has similar machinery, fine structural and functional aspects of each
synapse may dictate significantly the type of statistical transition [34].
4 Discussion
Applications of nonextensive statistics are not documented in brain synapses, although they
have been reported in many other systems. To address this issue, motivated by a limitation
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46 A. J. da Silva et al.
10 010 110 2
Frequency (Hz)
10 -1
10 0
VPSP (normalized)
100 200
Frequency (Hz)
0
0.5
1
VPSP (normalized)
Fig. 4 Electrophysiological data recorded with the extracellular field potential technique. Fitting of the
excitatory field VPSP of hippocampal synapse adapted from Fig. 1b of [23]. Full lines represent fitting with
(14) whereas dashed lines were fitted with (10). The inset shows the linear-linear representation of data. The
vertical line on the inset marks the crossover frequency at 16.026 Hz. The first crossover at 0.182 Hz is not
visible
of the 1/f -model to describe STD, we are proposing a new theoretical approach to reveal
both nonextensivity and possible statistical transition embedded in this type of plasticity.
Influenced by a paradigmatic nonextensive differential equation, we developed a nonlinear
model to be applied to electrophysiological recordings from brain synapses. In contrast to
the 1/f -model, our results agree with empirical data, providing a better adjustment in the
higher frequency stimulation range. A remarkable advantage of our proposal is the simplicity
to test long-range correlations and statistical crossover associated with the STD phenomenon.
Moreover, the results provide additional support against a synaptic transmission purely ruled
by random mechanisms. Although we recognized that our model still represents an oversim-
plified description of STD mechanisms, it preserves a manner to study long-range correla-
tions neglected in other theoretical descriptions. Therefore, a complete model, allowing a
rigorous conjunction of theoretical construction and physiological mechanisms, is still re-
quired to permit verification of nonextensivity in a more realistic physiological environment.
In this sense, calcium contribution, vesicle replenishment, and receptor desensitization are
fundamental requirements to expand this seminal model to a general STD description.
The obtained q-indexes values did not exhibit the confined range of 1 ≤q<3as
reported in our study carried out at the neuromuscular junction [13]. In this previous work,
Tab l e 1 Summary of simulations with best-fitted parameters obtained using (10), (14), (17), and (18). Time
constants, τwere extracted from [5,20,21,23,35]. Quantal sizes, Qfrom [16,36,37]
Reference q λ(s−1)μ(s−1)r τ(s) κ(s−1)Q(arb.unit)p
Fig. 3, Top 5.192 3.989 0.022 5.192 4.2 0.001 40.0 0.416
Fig. 3, Bottom 4.326 0.205 0.004 1.000 1.1 0.018 36.5 0.006
Fig. 47.933 0.790 0.009 1.013 8.0 0.001 40.0 0.156
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Statistical crossover and nonextensive behavior of the neuronal... 47
we adopted a q-Gaussian distribution in order to verify the presence of nonextensivity from
spontaneous miniature potentials. We used a q-Gaussian distribution that admits q-index
values within the range of 1 ≤q<3. However, q-exponential and r-exponential func-
tions do not require such restriction. For instance, q>3 was evaluated in psychophysical
data, image analysis, perceptual computing, and detection and location of mean level-shifts
in noise [38–41]. In our case, we attribute the existence of q>3orr>3 to a relaxation
process already suggested in studies involving stock markets and solar winds at the distant
heliosphere, respectively [42–44]. In fact, high values for the q-index are associated with a
relaxation process corresponding to a metastable state. In this sense, we hypothesized that
high q-index values arise due to statistical transitions and relaxation in the STD mechanism.
During stimulus application, vesicles placed at AZ are found to be in a balance between a
stable and an unstable state. In this sense, Long et al. [45] suggested a metastable state regu-
lating the vesicle fusion into a hemifusion process. In this framework, specific proteins may
also participate in this mechanism from the SNARE complex as indicated by Tang et al. [46].
The synaptic ending is a propitious biological system to observe the existence of nonex-
tensivity due to its peculiar ultrastructural features [47]. For instance, at the calyx of Held,
theAZareais0.1μm2, with two docked vesicles per AZ in a terminal volume of 480 μm3,
while at the hippocampal button, the AZ area is 0.039 μm2with ten docked vesicles per AZ
in a terminal volume of 0.08 μm3[48]. From such morphology, one may presume that a
smaller volume and a higher number of docked vesicles in hippocampal synapses constitute
physiological substrates consistent with higher q-indexes, as compared to giant auditory
synapses. Indeed, in restricted spatial dimensions, vesicle fusion on AZs can influence the
remaining vesicles to obtain a probability to be dragged in a multiquantal release or even
inhibiting the nearest vesicle to fuse with the terminal [49]. During the early stimulation
phase, the readily releasable pool is mobilized. However, further exocytosis, in response to
sustained stimulus, leads to depletion of the readily available pool and recruitment of the
other pools of vesicles. This non-uniform or heterogeneous neurotransmission is also sup-
ported by evidence of physical interactions among vesicles on the same AZ STD [50]. A
theoretical explanation for the heterogeneous framework for STD was achieved by Trom-
mershauser et al. assuming two pclasses. They associated a high pto the readily releasable
vesicles, released during the early stimulus, and low p, for those fusing at higher stimulation
levels [35]. These assumptions led the authors to study previous experiments from Gersdoff
et al., whose theoretical modeling are in agreement with empirical results [20]. However,
they do not consider physical interactions between vesicles as a source for a heterogeneous
STD mechanism.
In the present work, we suggest nonextensivity and statistical crossover as important
factors to explain STD heterogeneity. Heterogenous synapses, represented by statistical
transitions, guarantee fidelity over the transmission of a broad range of stimulation with-
out abolishing the postsynaptic response. Since q>1 reflects fractality, our results show
that STD presents a fractal behavior not previously described in other reports. A correspon-
dent physiological environment for r= 1 in auditory giant synapses and the hippocampus
can be interpreted using the morphological argument discussed above. As is well known,
higher frequencies promote a decrement of pby exhaustion of the readily releasable pool,
accelerating the recruitment of vesicles from other storages. If we consider that electrical
stimulations promote neural swelling, vesicle traffic facilitation is expected from these stor-
ages due to the increment of the intracellular milieu size [51]. High-frequency stimulus can
also accelerate the metabolism, decreasing the physical interaction likelihood on each AZ.
Combined, both aspects are arguments for a transition from a nonextensive to an exten-
sive behavior. Altogether, we advocate that, despite similarities in exocytosis mechanisms
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48 A. J. da Silva et al.
shared by different synapses, structural and functional elements inherent to each terminal
reflect STD statistical properties and nonextensivity degree.
5 Conclusions
To the best of our knowledge, this is the first work that assesses a nonextensive behavior
in brain synapses. In this framework, our main concern was formulating a physiological
model to uncover nonextensivity and possible statistical crossover in synaptic transmis-
sion. Despite functional and morphological particularities involved in each synapse here
studied, they share common features in their exocytosis machinery dynamics such as a sim-
ilar electrophysiological response. Both remarks encourage us to study how ubiquitous the
nonextensivity is in synapses by examining three different brain regions. We found a consis-
tent nonextensive scenario for mammalian and non-mammalian synapses, different species,
brain areas, and intracellular as well as extracellular compartments. In fact, the results val-
idated the Tsallis theory, at least in auditory and cortical neurons, evidencing that synaptic
transmission is not governed only by random mechanisms. Beyond that, statistical transi-
tions arising as a function of stimulus level provide novel and additional evidence in favor of
statistical heterogeneity in neurotransmitter release. Altogether, these findings represent an
important step toward the elaboration of more realistic models of STD mechanisms based
on nonextensive formalism. They also reinforce the richness and complexity of neuroplas-
ticity phenomena. Lastly, we hope elaborating experimental protocols for the acquisition
of our own data for further tests in a rigorous model taking into account a more realistic
physiological scenario.
Acknowledgments The authors thank Constantino Tsallis for his valuable suggestions and discussions.
Compliance with Ethical Standards
Conflict of interest Theauthorsdeclarethat there are no conflicts of interest associated with this publication.
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