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神經科學中的數學(導讀版)（簡體書）

系列名：神經科學研究與進展
ISBN13：9787030331144
出版社：科學出版社
作者：Fabrizio Gabbiani
出版日：2012/01/10
裝訂：精裝
規格：26cm*19cm (高/寬)
關鍵字：神經科學中的數學(導讀版)（簡體書）、神經、神經科、科學、數學、導讀、簡體、科學出版社、 Fabrizio Gabbiani、簡體書、數理科學和化學、

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數學

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《神經科學研究與進展:神經科學中的數學(英文)(導讀版)》通過Matlab編程語言在眾多模擬中的應用來介紹計算方法。這些程序為新的課程和研究提供有益的跳板。作者從介紹微分方程和線性代數在細胞、亞細胞和突起模型的應用開始，然后介紹概率論在突觸傳遞和單細胞噪聲中的應用，最后將信號處理理論應用于系統神經科學中。神經科學依賴眾多數學工具表達已有的理論、分析數據并提出新的實驗。《神經科學研究與進展:神經科學中的數學(英文)(導讀版)》采用一系列扎實的計算模型將該領域最令人矚目的工具由淺入深地介紹給讀者。旨在為神經科學專業的本科生和研究生，以及對神經科學感興趣的數學、物理和工程背景的學生提供一本教科書，亦可為進行神經科學相關研究的工作者提供有用的參考。

名人/編輯推薦

在神經科學的研究中，大量數學方法被用于表述已知原理和分析實驗數據。《神經科學研究與進展:神經科學中的數學(英文)(導讀版)》廣泛闡述了數學、神經科學及二者交叉領域的基礎原理，引導初學者逐步理解和掌握數學方法在神經科學中的應用。各章節交替介紹數學概念和數值計算方法，以及這些概念和方法在神經科學研究中的應用實例，涉及領域涵蓋傳統的細胞生物物理學研究和系統水平的神經生物學研究。數學知識循序漸進，深度遞增，便于非數學專業的讀者理解掌握。每章附有大量練習題及解答，內容均衍生于撰稿人的課程教材。每種計算模型包括MATLAB編碼，方便讀者使用。《神經科學研究與進展:神經科學中的數學(英文)(導讀版)》屬于通用工具，可以幫助神經生物學研究者采用合理的數據處理和統計方法得出結論。簡明易懂，適合初學者和相關領域的研究人員作為參考書。
《神經科學研究與進展:神經科學中的數學(英文)(導讀版)》特點：對線性代數、傅立葉變換、常微分和偏微分方程、概率論、隨機過程和信號處理理論進行數學和計算方面的系統介紹，闡述亞細胞、細胞和系統神經科學水平等多課題的基礎，大暈的練習題和解決方案輔助網站提供232個模擬結果的Matlab源代碼。

前言
1.簡介
1.1　如何使用本書
1.2　大腦簡介
1.3　基礎數學知識
1.4　計量單位
1.5　參考資料來源
2.被動等電位細胞
2.1　簡介
2.2　能斯特電位
2.3　膜電導
2.4　膜電容與電流平衡
2.5　突觸電導
2.6　總結與參考資料來源
2.7　練習題

前言
1.簡介
1.1　如何使用本書
1.2　大腦簡介
1.3　基礎數學知識
1.4　計量單位
1.5　參考資料來源
2.被動等電位細胞
2.1　簡介
2.2　能斯特電位
2.3　膜電導
2.4　膜電容與電流平衡
2.5　突觸電導
2.6　總結與參考資料來源
2.7　練習題
3.微分方程
3.1　精確求解
3.2　矩量法
3.3　拉普拉斯變換
3.4　數值分析法
3.5　突觸輸入
3.6　總結與參考資料來源
3.7　練習題
4.主動等電位細胞
4.1　延遲整流鉀離子通道
4.2　鈉離子通道
4.3　Hodgkin-Huxley公式
4.4　瞬時鉀離子通道
4.5　總結與參考資料來源
4.6　練習題
5.半主動等電位細胞
5.1　半主動模型
5.2　數值分析法
5.3　特征向量擴展的精確求解
5.4　持續鈉電流
5.5　超極化激活的非特異性陽離子電流
5.6　總結與參考資料來源
5.7　練習題
6.被動電纜理論
6.1　離散的被動電纜公式
6.2　特征向量擴展的精確求解
6.3　數值分析法
6.4　被動電纜公式
6.5　突觸輸入
6.6　總結與參考資料來源
6.7　練習題
7.傅立葉級數與變換
7.1　傅立葉級數
7.2　離散的傅立葉變換
7.3　連續傅立葉變換
7.4　協調離散與連續傅立葉變換
7.5　總結與參考資料來源
7.6　練習題
8.被動樹突野
8.1　離散的被動樹突野
8.2　特征向量擴展
8.3　數值分析法
8.4　被動樹突公式
8.5　（樹突電纜的）等效圓柱體
8.6　（樹突）分支的特征函數
8.7　總結與參考資料來源
8.8　練習題
9.主動樹突野
9.1　主動的均一電纜
9.2　主動均一電纜的相互作用
9.3　主動非均一電纜
9.4　半主動電纜
9.5　主動樹突野
9.6　總結與參考資料來源
9.7　練習題
10.簡化的單神經元模型
10.1　漏電的整合一發放神經元
10.2　簇放電神經元
10.3　簇放電神經元的簡化模型
10.4　總結與參考資料來源
10.5　練習題
11.概率與隨機變量
11.1　事件與隨機變量
11.2　符合二項式分布的隨機變量
11.3　符合泊松分布的隨機變量
11.4　符合高斯分布的隨機變量
11.5　累計分布函數
11.6　條件概率
11.7　獨立隨機變量的加和
11.8　隨機變量的變換
11.9　隨機向量
11.10　指數與伽瑪分布的隨機變量
11.11　齊次泊松過程
11.12　總結與參考資料來源
11.13　練習題
12.突觸傳遞與量子釋放理論
12.1　突觸的基本結構與生理
12.2　量子釋放的發現
12.3　突觸釋放的復合泊松模型
12.4　與實驗數據的對比
12.5　中樞系統突觸的量子分析
12.6　突觸傳遞的易化、增強與抑制
12.7　短時程突觸可塑性的模型
12.8　總結與參考資料來源
12.9　練習題
13.神經元鈣流信號
13.1　電壓門控型鈣離子通道
13.2　胞內鈣離子的擴散、緩釋與再攝取
13.3　電鏡所揭示的鈣離子釋放
13.4　樹突小棘的鈣離子
13.5　突觸前鈣離子和神經遞質釋放
13.6　總結與參考資料來源
13.7　練習題
14.奇異值分解算法及其應用
14.1　奇異值分解算法
14.2　主成分分析與動作電位發放的歸類
14.3　突觸可塑性與主成分
14.4　通過均衡截斷方法精簡神經元模型
14.5　總結與參考資料來源
14.6　練習題
15.動作電位發放波動的定量分析
15.1　動作電位發放的時間間距柱形統計圖與變異系數
15.2　動作電位不應期
15.3　動作電位發放數分布與法諾因子
15.4　更新過程
15.5　返回圖與經驗相關因子
15.6　總結與參考資料來源
15.7　練習題
16.隨機過程
16.1　定義與一般特性
16.2　高斯過程
16.3　點過程
16.4　非均一泊松過程
16.5　頻譜分析
16.6　總結與參考資料來源
16.7　練習題
17.膜噪聲
17.1　兩狀態通道模型
17.2　多狀態通道模型
17.3　Ornstein-Uhlenbeck過程
17.4　突觸噪聲
17.5　總結與參考資料來源
17.6　練習題
18.能量譜與互相關分析
18.1　互相關與互相干
18.2　估計量偏倚與方差
18.3　能量譜的數值估計
18.4　總結與參考資料來源
18.5　練習題
19.自然光信號與光（電）轉導
19.1　波長與光強
19.2　自然光信號的空間特性
……
20.（動作電位）發放率編碼與早期視覺
21.簡單細胞與復雜細胞模型
22.隨機估計理論
23.反相關分析與動作電位發放解碼
24.信號檢測理論
25.神經元反應與心理物理學的關聯研究
26.群體編碼
27.神經元網絡
28.練習題解答
參考文獻
索引

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書摘/試閱

1
 Introduction
O U T L I N E
1.1 How to Use this Book 2
1.2 Brain Facts Brief 2
1.3 Mathematical Preliminaries 4
1.4 Units 7
1.5 Sources 8
Faced with the seemingly limitless qualities of the
brain,Neuroscience has eschewed provincialism and instead
pursued a broad tack that openly draws on insights
frombiology,physics,chemistry,psychology,and mathematics in
its construction of technologies and theories with which to
probe and understand the brain.These technologies and
theories,in turn,continue to attract scientists and
mathematicians to questions of Neuroscience.As a result,we may
trace over one hundred years of fruitful interplay between
Neuroscience and mathematics.This text aims to prepare
the advanced undergraduate or beginning graduate student to take
an active part in this dialogue via the application
of existing,or the creation of new,mathematics in the interest
of a deeper understanding of the brain.Requiring no
more than one year of Calculus,and no prior exposure to
Neuroscience,we prepare the student by
1.introducing mathematical and computational tools in precisely
the contexts that first established their importance
for Neuroscience and
2.developing these tools in concrete incremental steps within a
common computational environment.
As such,the text may also serve to introduce Neuroscience to
readers with a mathematical and/or computational
background.
Regarding (1),we introduce ordinary differential equations via
the work of Hodgkin and Huxley (1952) on action
potentials in the squid giant axon,partialdifferential equations
throughthework ofRall oncable theory (see Segev et al.
(1994)),probability theory following the analysis of Fatt and
Katz (1952) on synaptic transmission,dynamical systems
theory in the context of Fitzhughs (1955) investigation of
action potential threshold,and linear algebra in the context
of the work of Hodgkin and Huxley (1952) on subthreshold
oscillations and the compartmental modeling of Hines
(1984) on dendritic trees.In addition,we apply Fourier
transforms to describe neuronal receptive fields following
Enroth-Cugell and Robsons (1966) work on retinal ganglion cells
and its subsequent extension toHubel andWiesels
(1962) characterization of cat cortical neurons.We also
introduce and motivate statistical decision methods starting
with the historical photon detection experiments of Hecht et
al.(1942).
Regarding (2),we develop,test,and integratemodels of
channels,receptors,membranes,cells,circuits and sensory
stimuli by working from the simple to the complex within the
MATLAB computing environment.Assuming no prior
exposure toMATLAB,we develop and implement numerical methods for
solving algebraic and differential equations,
for computing Fourier transforms,and for generating and
analyzing random signals.Through an associatedweb site
we provide the studentwithMATLAB code for 144
computationalfigures in the text andwe provide the instructor
with
MATLAB code for 98 computational exercises.The exercises range
from routine reinforcement of concepts developed
in the text to significant extensions that guide the reader to
the research literature.Our reference to exercises both in
the text and across the exercises serve to establish them as an
integral component of this book.
Concerning themathematical models considered in the text,we cite
the realization of Schr?dinger (1961) that "we
cannot ask for more than just adequate pictures capable of
synthesizing in a comprehensible way all observed facts
and giving a reasonable expectation on new oneswe are out
for."Furthermore,lest "adequate"serve as an invitation
to loose or vague modeling,Schr?dinger warns that "without an
absolutely precise model,thinking itself becomes
imprecise,and the consequences derived from the model become
ambiguous.”
As we enter the 21st century,one of the biggest challenges
facing Neuroscience is to integrate knowledge and to
craft theories that span multiple scales,both in space from the
nanometer neighborhood of an ion channel to the
meter that action potentialsmust travel down the sciatic
nerve,and in time from the fraction of a millisecond it takes
to release neurotransmitter to the hours it takes to prune or
grow synaptic contacts between cells.We hope that this
text,by providing an integrated treatment of experimental and
mathematical tools within a single computational
framework,will prepare our readers to meet this challenge.
1.1 HOW TO USE THIS BOOK
The book is largely self-contained and as such is suited for
both self-study and reference use.The chapters need not
be read in numerical order.To facilitate a selection for
reading,we have sketched in Figure 1.1 themain dependencies
between the chapters.The four core chapters that underlie much
of the book are Chapters 2-4 and 11.For the reader
with limited prior training in mathematics it is in these
chapters that we develop,by hand calculation,MATLAB
simulation and a thorough suite of exercises,the mathematical
maturity required to appreciate the chapters to come.
Many of the basic chapters also contain more advanced
subsections,indicated by an asterisk,?,which can be skipped
on a first reading.Detailed solutions are provided for most
exercises,either at the end of the book or through the
associatedweb site.We mark with a dagger,?,each exercise whose
solution is not included in this text.
Over the past eight years,we have used a subset of the books
material for a one semester introductory course
on Mathematical Neuroscience to an audience comprised of Science
and Engineering undergraduate and graduate
students from Rice University and Neuroscience graduate students
from Baylor College of Medicine.We first cover
Chapters 2-5,which set and solve the Hodgkin-Huxley equations
for isopotential cells and,via the eigenvector
expansion of the cells subthreshold response,introduce the key
concepts of linear algebra needed to tackle the
multicompartment cell in Chapters 6 and 8-9.We then open Chapter
11,introduce probabilistic methods and apply
them to synaptic transmission,in Chapter 12,and spike train
variability,in Chapter 15.We conclude this overview
of single neuron properties by covering Chapter 10 on reduced
single neuron models.We transition to Systems
Neuroscience via the Fourier transform of Chapter 7 and its
application to visual neurons in Chapters 20 and 21.
Finally,we connect neural response to behavior via thematerial
of Chapters 24 and 25.An alternative possibility is to
conclude withChapters 22 and 23,after an informal introduction
to stochastic processes,and power and cross spectra
in Chapters 16 and 18.
We have also used the following chapters for advanced courses:
13,14,16-19,and 26.Chapter 13 provides a
comprehensive coverage of calcium dynamics within single neurons
at an advanced level.Similarly,Chapter 14
introduces the singular value decomposition,a mathematical tool
that has important applications both in spike
sorting and in model reduction.Chapters 16 and 18 introduce
stochastic processes and methods of spectral analysis.
These results can be applied at the microscopic level to
describe single channel gating properties,Chapter 17,and at
the macroscopic level to describe the statistical properties of
natural scenes and their impact on visual processing,
Chapter 19.Finally the chapters on population codes and
networks,Chapters 26 and 27,address the coding and
dynamical properties of neuronal ensembles.
To ease the reading of the text,we have relegated all references
to the Summary and Sources section located at the
end of each chapter.These reference lists are offered as
pointers to the literature and are not intended to be
exhaustive.
1.2 BRAIN FACTS BRIEF
The brain is the central component of the nervous system and is
incredibly varied across animals.In vertebrates,it
is composed of threemain subdivisions: the
forebrain,themidbrain,and the hindbrain.Inmammals and
particularly
in humans,the cerebral cortex of the forebrain is highly
expanded.The human brain is thought to contain on the
order of 100 billion (1011) nerve cells,or neurons.Each neuron
"typically"receives 10,000 inputs (synapses,§2.1)
from other neurons,but this number varies widely across neuron
types.For example: granule cells of the cerebellum,
the most abundant neurons in the brain,receive on average four
inputs while Purkinje cells,the output neurons of
the cerebellar cortex,receive on the order of 100,000.In the
mouse cerebral cortex,the number of neurons per cubic
millimeter has been estimated at 105,while there are
approximately 7×108 synapses and 4 km of cable (axons,§2.1)
in the same volume.Brain size (weight) typically scales with
body size,thus the human brain is far from the largest.
At another extreme,the brain of the honey bee is estimated to
contain less than a million (106) neurons within a
single cubic millimeter.Yet the honey bee can learn a variety of
complex tasks,not unlike those learned by a macaque
monkey for instance.Although it is often difficult to draw
comparisons across widely different species,the basic
principles underlying information processing as they are
discussed in this book appear to be universal,in spite of
obvious differences in implementation.The electrical properties
of cells (Chapter 2),the generation and propagation
of signals along axons (Chapters 4 and 9),and the detection of
visual motion (Chapters 21 and 25) or population codes
(Chapter 26),for instance,are observed to follow very similar
principles across very distantly related species.
Information about the environment reaches the brain through five
common senses: vision,touch,hearing,smell,
and taste.In addition,some animals are able to sense electric
fields through specialized electroreceptors.These include
many species of fish andmonotremes (egg-layingmammals) like the
platypus.Most sensory information is gathered
from the environment passively,but some species are able to emit
signals and register their perturbation by the
environment and thus possess active sensory systems.This
includes bats that emit sounds at specific frequencies and
hear the echoes bouncing off objects in the environment,a
phenomenon called echolocation.In addition some species
of fish,termedweakly electric,possess an electric organ allowing
them to generate an electric field around their body
and sense its distortion by the environment,a phenomenon called
electrolocation.
Ultimately,the brain controls the locomotor output of the
organism.This is typically a complex process,involving
both commands issued to the muscles to execute
movements,feedback from sensors reporting the actual state of
the
musculature and skeletal elements,and inputs from the senses to
monitor progress towards a goal.So efficient is this
process that even the tiny brain of a fly is,for instance,able
to process information sufficiently fast to allow for highly
acrobatic flight behaviors,executed in less than 100 ms from
sensory transduction to motor output.
To study the brain,different biological systems have proven
useful for different purposes.For example,slices of the
rat hippocampus,a structure involved in learning and memory as
well as navigation,are particularly adequate for
electrophysiological recordings of pyramidal neurons and a
detailed characterization of their subcellular properties,
because their cell bodies are tightly packed in a layer that is
easy to visualize.The fruit fly Drosophila melanogaster and
thewormCaenorhabditis elegans (whosenervous systemcomprises
exactly 302 neurons) are goodmodels to investigate
the relation between simple behaviors and genetics,as their
genomes are sequenced and many tools are available to
selectively switch on and off genes in specific brain structures
or neurons.One approach that has been particularly
successful to study informationprocessing in the brain is
"neuro-ethological,"based on the study of natural behaviors
(ethology) in relation to the brain structures involved in their
execution.Besides the alreadymentionedweakly electric
fish and bats,classical examples,among many others,include song
learning in zebrafinches,the neural control of
flight in flies,sound localization in barn owls,and escape
behaviors in a variety of species,such as locust,goldfish,
or flies.
1.3 MATHEMATICAL PRELIMINARIES
MATLAB.Native MATLAB functions are in typewriter
font,e.g.,svd.Our contributed code,available on the books
web site,has a trailing .m,e.g.,bepswI.m.
Numbers.The counting numbers,{0,1,2,...},are denoted by N,while
the reals are denoted by R and the complex
numbers by C.Each complex number,z∈C,may be decomposed into its
real and imaginary components.Wewillwrite
z=x+iy,where x=(z),y=(z),and i ≡ √?1.
Here x and y are each real and ≡ signifies that one side is
defined by the other.We denote the complex conjugate and
magnitude of z by
z? ≡x?iy and |z|≡x2+y2,
respectively.
Sets.Sets are delimited by curly brackets,{}.For example the set
of odd numbers between 4 and 10 is {5,7,9}.
Intervals.For a,b∈R with a
[a,b] is the set of numbers x such that a≤x≤b.The semiclosed (or
semiopen) intervals [a,b) and (a,b] are the set of
numbers x such that a≤x
Vectors and matrices.Given n real or complex
numbers,x1,x2,...,xn,we denote their arrangement into a
vector,or
column,via bold lower case letters,
x=x1x2...xn
.(1.1)
The collections of all real and complex vectors with n
components are denoted Rn and Cn,respectively.The transpose
of a vector,x,is the row,
xT =(x1 x2…xn),
and the conjugate transpose of a vector,z∈Cn,is the row
zH =(z?1 z?2…z?n).
We next define the inner,or scalar,or "dot,"product for x and y
in Cn,
xHy≡nj=1
x?j yj,and note that as
zHz=i=1
|zi|2 ≥0
MATHEMATICS FOR NEUROSCIENTISTS
it makes sense to define the norm
 
z
≡
zHz.
To gain practice with these definitions you may wish to confirm
that
 
(yHy)x?(yHx)y
2=
y
2(
y
2
x
2?|xHy|2).
As the left hand side is nonnegative the right hand side reveals
the important Schwarz inequality
|xHy|≤
x
 
y
.(1.2)
We will transform vectors in Cn to vectors in Cm via
multiplication by m×n matrices,A∈Cm×n,of the form
A=?????
A11 A12…A1n
A21 A22…A2n
Am1 Am2…Amn
?????
where each Aij ∈C.Thus,y=Ax means that yi =nj
=1Aijxj for i=1,...m.Wewill consistently denotematrices by
bold
upper case letters.Given A∈Cm×n and B∈Cn×p we define their
product C∈Cm×p via
C=AB where Cjk =
n 
l=1
AjlBlk .
If we reflect A about its diagonal we arrive at its transpose,AT
∈Cn×m,where (AT)ij =Aji.The associated conjugate
transpose is denoted AH,where (AH)ij =A?ji.We will often require
the conjugate transpose of the product AB,and so
record
(AB)H
jk =nl=1
A?klB?lj =(BHAH)jk,i.e.,(AB)H =BHAH.(1.3)
Similarly,(AB)T =BTAT.
The identity matrix,denoted I,is the square matrix of zeros off
the diagonal,and ones on the diagonal,
We often use the Kronecker delta
δjk ≡1,if j=k
0,otherwise
(1.4)
to denote the elements of I.Amatrix B∈Cn×n is said to be
invertible if there exists a matrix B?1 ∈Cn×n such that
BB?1=B?1B=I.(1.5)
In this case B?1 is called the inverse of B.
MATHEMATICS FOR NEUROSCIENTISTS
Functions.We will make frequent use of the characteristic
function
1(a,b)(x)≡1,if a<x<b
0,otherwise
(1.6)
of the interval,(a,b).In the common case that (a,b) is the set
of nonnegative reals we will simply write 1(x) and refer
to it as the Heaviside function.
We will often need to differentiate the running integral,
F(x)=
x
0f (y)dy.
To see that
F(x)=f (x) (1.7)
when f is continuous at x,note that the mean value theorem
establishes the second equality in
F(x+h)?F(x)
h =1hx+h
x
f (y)dy=f (xh) (1.8)
for some xh ∈(x,x+h).As h→0 the left hand side approaches F(x)
while on the right xh→x and so,by continuity,
f (xh)→f (x).
We will often need to sample,or discretize,scalar valued
functions,f :R→R,of time and/or space.For example,
if time is divided into increments of size dt then we will
denote the samples of f (t) by superscripted letters in the
"typewriter"font
fj ≡f (( j?1)dt),j=1,2,3,....
Similarly,we will denote the samples of a vector valued
function,f :R→Rn,by superscripted letters in the bold
typewriter font
fj ≡f(( j?1)dt),j=1,2,3,....
The elements of fj are samples of the elements of f.We express
this in symbols as fj
m=fm(( j?1)dt).Where the
superscript,j,may interfere with exponents we will be careful to
make the distinction.
Random variables.In chapters dealing with random variables,we
will try whenever possible to use upper case
letters for a random variable and lower case letters for a
specific value of the same random variable.We denote the
expectation or mean of a random variable X by E[X].The variance
is the expectation of the squared deviation of
X from the mean: E[(X?E[X])2].A Gaussian or normal random
variable of mean μ and variance σ2 is denoted by
N(μ,σ2 ).An estimator of,e.g.,the mean mX of a random variable X
is denoted by m? X.
Fourier transforms.The Fourier transform of a function f (t) of
time,t,is denoted by ?f (ω):
?f (ω)≡
∞
 ?∞
f (t)e?2πiωtdt.
The variable ω is the ordinary frequency.If t has units of
seconds (s) then ω has units of 1/s=Hz (Hertz).The
convolution of two functions f and g is denoted by fg:
( f g)(t)≡
∞?∞
f (t1)g (t?t1)dt1.(1.9)

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神經科學中的數學(導讀版)（簡體書）

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系列名：神經科學研究與進展

ISBN13：9787030331144

出版社：科學出版社

作者：Fabrizio Gabbiani

出版日：2012/01/10

裝訂：精裝

規格：26cm*19cm (高/寬)

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