理论神经科学：研究生版

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"Theoretical Neuroscience for Graduate-Edition4 .pdf" 本书是针对研究生级别的理论神经科学教材，由Peter Dayan和L.F. Abbott撰写。它深入探讨了神经元响应的分析和建模方法，旨在帮助读者理解大脑如何处理信息。在第一章“神经编码I：放电率与尖峰统计”中，作者首先介绍了神经元的基本属性，如它们是如何记录和传递信息的。神经元响应通常通过记录其放电活动（尖峰）来研究，从刺激到响应的转换过程被详细阐述。放电率是衡量神经元活动强度的一个关键指标，用于描述神经元对特定刺激的反应程度。调谐曲线则展示了神经元对不同刺激的敏感度。尖峰计数变异性是另一个重要的概念，它讨论了神经元放电模式的不稳定性。接下来，作者探讨了什么因素能触发神经元放电，以及如何通过描述刺激来理解这一过程。其中，尖峰触发平均值（Spike-Triggered Average）被用来揭示神经元对特定刺激片段的反应模式。白噪声刺激被广泛使用，因为它可以提供丰富的刺激信息。此外，还讨论了多个尖峰触发平均值和尖峰触发相关性，这些对于理解神经元间的交互作用至关重要。在尖峰序列统计部分，作者介绍了齐次泊松过程和非齐次泊松过程，以及它们在模拟神经放电中的应用。这些统计模型与实际神经数据进行比较，以评估其适用性。书中还探讨了神经编码的不同类型，如独立尖峰、独立神经元编码和相关编码，以及时间编码的概念，这些编码方式都反映了神经信息传输的复杂性。第二章“神经编码II：反向相关和感受野”进一步深入，介绍了一种估计放电率的方法——反向相关法，这是理解神经元感受野（Receptive Fields）的重要工具。反向相关法可以帮助研究人员识别神经元对输入信号的响应特征，从而揭示神经元如何编码视觉、听觉等感官信息。这本书涵盖了神经科学中从基本的神经元特性到高级的编码策略等多个层次的内容，结合数学模型和统计方法，为研究生提供了全面理解神经信息处理的基础。书中还包含了丰富的附录和参考文献，以便于读者进一步探索相关主题。

6 Neural Encoding I: Firing Rates and Spike Statistics

ing, is completely absent on the axon due to attenuation, but the action

potential sequence in the two recordings is the same. This illustrates the

important point that spikes, but not subthreshold potentials, propagate

regeneratively down axons.

Themiddletraceinﬁgure 1.3 illustrates an idealized, noise-free extracel-

lular recording. Here an electrode is placed near a neuron but it does not

penetrate the cell membrane. Such recordings can reveal the action poten-extracellular

electrodes tials ﬁred by a neuron, but not its subthreshold membrane potentials. Ex-

tracellular recordings are typically used for in vivo experiments, especially

those involving behaving animals. Intracellular recordings are sometimes

made in vivo, but are more commonly used for in vitro preparations such

as experiments on slices of neural tissue. The responses studied in this

chapter are action potential sequences that can be recorded either intra- or

extra-cellularly.

From Stimulus to Response

Characterizing the relationship between stimulus and response is difﬁcult

because neuronal responses are complex and variable. Neurons typically

respondby producingcomplex spike sequences that reﬂect both the intrin-

sic dynamics of the neuron and the temporal characteristics of the stimu-

lus. Isolating features of the response that encode changes in the stimulus

canbedifﬁcult, especially if the time scale for these changes is of the same

order as the average interval between spikes. Neural responses can vary

from trial to trial even when the same stimulus is presented repeatedly.

There are many potential sources of this variability including variable lev-

els of arousal and attention, randomness associated with various biophys-

ical processes that affect neuronal ﬁring, and the effects of other cognitive

processestaking place during a trial. The complexity and trial-to-trial vari-

ability of action potential sequences make it unlikely that we can describe

and predict the timing of each spike deterministically. Instead, we seek a

model that can account for the probabilities that different spike sequences

are evoked by a speciﬁc stimulus.

Typically, many neurons respond to a given stimulus, and stimulus fea-

tures are therefore encoded by the activities of large neural populations. In

studying population coding, we must examine not only the ﬁring patterns

of individual neurons, but also the relationships of these ﬁring patterns to

each other across the population of responding cells.

In this chapter, we introduce the ﬁring rate and spike-train correlation

functions, which are basic measures of spiking probability and statistics.

We also discuss spike-triggered averaging, a method for relating action

potentials to the stimulus that evoked them. Finally, we present basic

stochastic descriptions of spike generation, the homogeneous and inho-

mogeneous Poisson models, and discuss a simple model of neural re-

sponses to which they lead. In chapter 2, we continue our discussion of

Peter Dayan and L.F. Abbott Draft: December 17, 2000

1.2 Spike Trains and Firing Rates 7

neural encoding by showing how reverse-correlation methods are used

to construct estimates of ﬁring rates in response to time-varying stimuli.

These methods have been applied extensively to neural responses in the

retina, lateral geniculate nucleus (LGN) of the thalamus, and primary vi-

sual cortex, and we review the resulting models.

1.2 Spike Trains and Firing Rates

Action potentials convey information through their timing. Although ac-

tion potentials can vary somewhat in duration, amplitude, and shape,

they are typically treated in neural encoding studies as identical stereo-

typed events. If we ignore the brief duration of an action potential (about

1 ms), an action potential sequence can be characterized simply by a list

of the times when spikes occurred. For n spikes, we denote these times

by t

with i

= 1,

2,... ,n. The trial during which the spikes are recorded

is taken to start at time zero and end at time T,so0

≤ t

≤ T for all i.The

spike sequence can also be represented as a sum of inﬁnitesimally narrow,

idealized spikes in the form of Dirac

δ functions (see the Mathematical

Appendix),

ρ(t) =



i=1

δ(t −t

). (1.1)

We call

ρ(t) the neural response function and use it to re-express sums neural response

function

ρ(t)over spikes as integrals over time. For example, for any well-behaved

function h

(t), we can write



i=1

h(t −t

) =



dτ h(τ)ρ(t −τ) (1.2)

where the integral is over the duration of the trial. The equality follows

from the basic deﬁning equation for a

δ function, δ function



τδ(t −τ)h(τ) = h(t), (1.3)

provided that the limits of the integral surround the point t (if they do not,

the integral is zero).

Because the sequence of action potentials generated by a given stimulus

typically varies from trial to trial, neuronal responses are typically treated

probabilistically, and characterized, for example, by the probability that a

spike occurs at a particular time during a trial. Spike times are continuous

variables, and, as a result, the probability for a spike to occur at any pre-

cisely speciﬁed time is actually zero. To get a nonzero value, we must ask

for the probability that a spike occurs within a speciﬁed interval, for exam-

ple the interval between times t and t

+t. For small t, the probability

Draft: December 17, 2000 Theoretical Neuroscience

8 Neural Encoding I: Firing Rates and Spike Statistics

of a spike falling in this interval is proportional to the size of the interval,

t. A similar relation holds for any continuous stochastic variable z.The

probability that z takes a value between z and z

+z, for small z (strictly

speaking, as

z → 0) is equal to p[z]z, where p[z] is called a probability

density. Throughout this book, we use the notation P[] to denote proba-

bilities and p[] to denote probability densities. We use the bracket nota-

tion, P[], generically for the probability of something occurring and also

to denote a speciﬁc probability function. In the latter case, the notation

() would be more appropriate, but switching between square brackets

and parentheses is confusing, so the reader will have to use the context to

distinguish between these cases.

For the particular case of spike occurrences, we can write the probability

that a spike occurs between times t and t

+ t, for small t as p[t]t,

where p[t] is the single spike probability density. The probability density

for the occurrence of a spike is, by deﬁnition, the ﬁring rate of the cell, and

we use the notation p[t]

= r(t) for this important quantity.ﬁring rate r(t)

The ﬁring rate at time t, r(t), can be estimated by determining the frac-

tion of trials with a given stimulus on which a spike occurred between the

times t and t

+t. For sufﬁciently small t and sufﬁciently large num-

bers of trials, this fraction provides a good estimate of r

(t), as guaranteed

by the law of large numbers. The fraction of trials on which a spike oc-

curs can be computed from the neural response function averaged over

trials. We use angle brackets,

, to denote averages over trials that usetrial average 

the same stimulus, so that z for any quantity z is the sum of the values

of z obtained from many trials involving the same stimulus, divided by

the number of trials. The trial-averaged neural response function is thus

denoted by

ρ(t). In any integral expression such as equation 1.2, the

neural response function generates a contribution whenever a spike oc-

curs. If instead, we use the trial-average response function in equation 1.2,

this generates contributions proportional to the fraction of trials on which

a spike occurred. Because of the relationship between this fraction and the

ﬁring rate, we ﬁnd that

(t)t =



t+t

dτ ρ(τ ). (1.4)

Furthermore, within any well-behaved integral, we can replace the trial-

averaged neural response function by the single-spike probability density

or ﬁring rate and write



τ h(τ)



ρ(t −τ)





τ h(τ)r(t −τ) (1.5)

for any function h. This establishes an important relationship between the

average neural response function and the ﬁring rate; the two are equiva-

lent when used inside integrals.

We call the single-spike probability density, r

(t),theﬁring rate. However,

this term is conventionally applied to more than one quantity. A differ-

Peter Dayan and L.F. Abbott Draft: December 17, 2000

1.2 Spike Trains and Firing Rates 9

ent ﬁring rate, which we call the spike-count rate, is obtained simply by

counting the number of action potentials that appear during a trial and

dividing by the duration of the trial. Unlike r

(t), the spike-count rate can

be determined for a single trial. We denote the spike-count rate by r (as spike-count rate r

opposed to r

(t) for the single-spike probability density) where



dτρ(τ).

(1.6)

The second equality follows from the fact that



τρ(τ)= n and indicates

that the spike-count rate is the time average of the neural response func-

tion over the duration of the trial.

In the same way that the response function

ρ(t) can be averaged across

trials to give the ﬁring rate r

(t), the spike-count ﬁring rate can be averaged

over trials yielding a quantity that we refer to as the average ﬁring rate.

This is denoted by

r and given by trial average

rate



r



r=



n



dτ ρ(τ )=



dtr(t). (1.7)

The third equality follows from the equivalence of the ﬁring rate and the

trial averaged neural response function within integrals, equation 1.5. The

average ﬁring rate is equal to both the time average of r

(t) and the trial

average of the spike-count rate r. Of course, a spike-count rate and average

ﬁring rate can be deﬁned by counting spikes over any time period, not

necessarily the entire duration of a trial.

The term ﬁring rate is commonly used for all three quantities, r(t), r, and

r.Weusethetermsﬁring rate, spike-count rate, and average ﬁring rate

for r

(t), r, and r respectively whenever possible but, when this becomes

too cumbersome, the different mathematical notations serve to distinguish

them. In particular, we distinguish the spike-count rate r from the single-

spike probability density r

(t) by using a different font and by including

the time argument in the latter expression (unless r

(t) is independent of

time). The difference between the fonts is rather subtle, but the context

should make it clear which rate is being used.

Measuring Firing Rates

The ﬁring rate r(t), being a probability density, cannot be determined ex-

actly from the limited amounts of data available from a ﬁnite number of

trials. In addition, there is no unique way to approximate r

(t). A discus-

sion of the different methods allows us to introduce the concept of a linear

ﬁlter and kernel that will be used extensively in the following chapters.

We illustrate these methods by extracting ﬁring rates from a single trial,

but more accurate results could be obtained by averaging over multiple

trials.

Draft: December 17, 2000 Theoretical Neuroscience

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