Teorie e modelli nelle scienze cognitive (parte...

Teorie e modelli nelle scienze cognitive (parte 1)

Corso di Principi e Modelli della Percezione

Prof. Giuseppe Boccignone

Dipartimento di InformaticaUniversità di Milano

[email protected]://boccignone.di.unimi.it/PMP_2018.html

Percezione: principi e livelli di spiegazione

Scienze cognitive

Psicologia cognitiva

Intelligenza artificiale

NeuroscienzeLinguistica

Filosofia della mente

Antropologia

Percezione

Percezione: modelli (teorie) computazionali

Scienze cognitive

Psicologia cognitiva

Intelligenza artificiale

NeuroscienzeLinguistica

Filosofia della mente

Antropologia

Percezione

Teoria computazionale

Rappresentazione e algoritmi

Implementazione fisica

Probabilità & statisticaAlgebra lineareSimulazioni

Teorie e modelli: filosofia della scienza //il modello induttivo

osservazioniesperimenti

teorie (= generalizzazioni)

predizioni

la predizione è confermata?

la teoria è più probabile

la teoriaè falsa

sì

no

Inferenza induttiva

Inferenza deduttiva


osservazioniesperimenti

teorie (= generalizzazioni)

predizioni



la teoriaè falsa

sì

no

Inferenza induttiva

Inferenza deduttiva

Cosa sono Teorie & Modelli

Il problema della falsificabilità


teorie

predizioni



la teoriaè falsa

sì

no

Inferenza induttiva

Inferenza deduttiva


• Le teorie scientifiche sono costituite da asserzioni universali (ipotesi, leggi, principi)

• In fisica: i principi di Newton, le leggi di Maxwell dell’elettrodinamica, i principi della termodinamica, i principi della meccanica quantistica, ecc

• In biologia: i principi dell’evoluzione

• Si arriva ad esse attraverso un processo di induzione, che parte da asserzioni singolari, cioè da resoconti dei risultati di osservazioni o esperimenti

• Problema (Hume): è giustificabile logicamente l'inferenza di asserzioni universali da asserzioni particolari, per quanto numerose queste siano?

• dal fatto che molti cigni sono bianchi non si può concludere che "tutti i cigni sono bianchi"

Teorie e modelli: filosofia della scienza //il modello ipotetico-deduttivo

• logica (predicativa) • teoria/osservazione • leggi e “regole di corrispondenza” • verificazione e significato empirico

la struttura di una teoria scientifica

teoria T1 ⇒ T2

-------⇑ ----- ⇓ ------ osservazione C → K

L1, L2, … (leggi universali)

C1, C2, … (condizioni iniziali) ----------------------------------- K

il modello “nomologico-deduttivo” della spiegazione e della previsione

spiegazione (K = descrizione di un fatto già noto)

previsione (K = descrizione di un fatto da accertare)

la crescita della scienza: “riduzione” e “accumulazione”

Galileo

KepleroNewton Einstein

Teorie e modelli: filosofia della scienza //la standard view • Una teoria scientifica si costituisce generalmente di tre elementi:

• Un vocabolario:

• Le proposizioni costitutive di una teoria scientifica sono rappresentate da

• (1) postulati [o assiomi o proposizioni/ipotesi fondamentali/primitive o leggi teoriche],

• (2) definizioni,

• (3) teoremi [o ipotesi/proposizioni derivative].

• Un sistema di regole di collegamento delle proposizioni teoriche alle proposizioni osservative espresse in forma di legge:

• regole semantiche (Carnap), definizioni coordinatrici (Reichenbach), definizioni operative (Bridgman), correlazioni epistemologiche (Margenau), regole di interpretazione (Northrop); e ancora, regole di trasformazione (Harré), principi ponte (Hempel), proposizioni interpretative (Hempel); proposizione di riduzione (Carnap)

• Un modello interpretativo della teoria.

• Hempel,

• "una teoria scientifica può essere paragonata a una complessa rete spaziale: i suoi termini sono rappresentati dai nodi, mentre i fili che li connettono corrispondono, in parte, alle definizioni e, in parte, alle ipotesi fondamentali e derivate incluse nella teoria. L'intero sistema naviga, per così dire, sopra il piano dell'osservazione ed è ancorato a quest'ultimo da regole di interpretazione. Queste possono essere viste come funi che non sono parte della rete ma collegano certi punti della rete a specifici luoghi del piano dell'osservazione

Teorie e modelli: filosofia della scienza //la standard view

Teorie e modelli: filosofia della scienza //Teorie vs modelli: due sensi di modello

• (1) Un modello in senso logico è qualunque struttura astratta che renda vero un insieme di assiomi (Suppes), in cui sia definita una funzione che fa corrispondere:

• a ogni costante e a ogni variabile del linguaggio un individuo nell’insieme O di oggetti che costituisce il dominio della teoria

• ad ogni predicato e relazione del linguaggio opportuni sottoinsiemi di O: per esempio i predicati del linguaggio individueranno il sottoinsieme di O tale che tutti i suoi membri soddisfano la proprietà cui corrisponde il predicato in questione (“è un gas”)

• alle relazioni binarie coppie di elementi


• (2) Rappresentazione semplificata e astratta dei fenomeni, definita dalle leggi di natura .

• Esempi:

• La prima legge del moto e il problema dell’attrito

• La legge della gravitazione universale, insieme alla seconda legge del moto, identificano un modello newtoniano di un sistema gravitazionale

• La legge della molla F = -kx identifica il modello del relativo sistema fisico.

• Un pendolo semplice è un modello del pendolo reale formato da un filo non-estensibile e non soggetto ad attrito, ma i fili reali sono tutti estensibili e soggetti sia all’attrito che alla resistenza dell’aria.

• L’astrazione da qualità e da proprietà causali reali


• Che rapporto esiste tra modello e realtà fisica?

• C’è un qualche tipo di somiglianza (Giere 1988)

• Modelli = Rappresentazioni

• Rappresentazione: S usa il modello M per rappresentare il mondo W per lo scopo P

• C’è un qualche tipo di isomorfismo, magari parziale (realismo strutturale)

• Non c’è alcun rapporto!

Teorie e modelli: filosofia della scienza //Modelli come rappresentazioni (Giere)

• Eterogeneità dei modelli:

• modelli teorico/formali

• modelli analogici

• modelli fisici

• modelli di scala

Teorie e modelli: filosofia della scienza //Le teorie come insiemi di modelli (Giere)

• I call my understanding of models representational because it takes models not primarily as providing a means for interpreting formal systems, but as tools for representing the world. That is not their only function, but it is, I think, the central function of models used in empirical science.

• On my interpretation, the model/theory distinction is mainly a reflection of the extent to which a branch of inquiry is guided by broad general principles.

• Where there are such principles, as in many areas of physics and biology, the models employed often, though not always, embody these principles. Where such principles are lacking, the models employed derive principally from various mathematical techniques. In both cases, however, reasoning about the world is primarily reasoning with models.

• It is models almost all the way up.

Modelli nelle scienze cognitive e nella percezione //complessità e livelli di spiegazione

Modelli nelle scienze cognitive e nella percezione //livelli di spiegazione


Qual è il goal della computazione?

Quale rappresentazione e quale algoritmo?

Come realizzarla fisicamente?

Livelli di spiegazione

secondo Marr


Teoria computazionale

Rappresentazione e algoritmo

Implementazione hardware


secondo Marr






secondo Marr






secondo Marr

Livello psicologico

Livello neurofisiologico






Livello psicologico

Livello neurofisiologico

Modello teorico

Simulazione





Modello teorico

Simulazione

Modello alLivello n

ImplementazioneLivello n

Bayesian Theory

Constraintsand

Hypotheses

Implementation Theory

T E

F

Affective state

Expressed features

Behavioural Theories &

Observations

NeuralTheories& Observations

Neuroimaging

25

• Modelli Bayesiani

Personal traits

A4Network 3Network 2

Nature Reviews | Neuroscience

Network 1A1 A3A2

Cognitive

Behaviours

Neuralcomputations

Brainareas

Affe

ctive

NC2 NC3 NC4NC1

and monitoring errors. For instance, ACC neurons encode the probability of reward120, and at the same time, are critical for making decisions about effort costs121–123. Thus, the ACC has a role in evaluating whether the benefits are worth the cost of making an action124. Second, the LPFC has a role in maintaining and manipulating infor-mation, but can also integrate this content with both affective and motivational infor-mation (BOX 3). The LPFC might therefore act as a control hub in which multiple types of information converge and are integrated (see below). Critically, the convergence of both cognitive and affective/motivational information enables the LPFC to dynami-cally weigh multiple types of information in guiding action. Finally, the above discussion has been simplified by treating control as a somewhat monolithic process. In reality, control probably involves a range of proc-esses that might vary from more ‘reactive’ to more ‘effortful’21 and that could have developmental trajectories125.

A conceptual proposalThe previous section briefly outlined the circuits involved in the evaluation of sensory information and executive control. The description of these circuits is by neces-sity incomplete, for the reasons described below and in several discussions and proposals89,126–133.

The simplest way to conceptualize the mapping between a brain area and behaviour is to assume a one-to-one map-ping between an area and its function. For instance, the V1 is linked to visual perception (or a set of more basic visual functions, such as edge detection and stereopsis). Even though such an exercise becomes considerably less straightforward for more central areas (that is, farther from the sensory periphery), we can imagine extending it to many areas of the brain. The end product of such a strategy would be a list of area–function pairs. The brain areas might then be labelled as cognitive or affec-tive based on their purported functions or how these functions are envisioned to ultimately shape behaviour. For instance, we could describe the amygdala as affective given its involvement in fear conditioning, and the ACC as cognitive given its role in monitoring conflict. As discussed, I believe that such dichotomization provides a poor description of the brain/mind, and I propose here that the one-area/ one-function viewpoint, although com-monly used (or at the very least implicitly assumed), is problematic.

Structural connectivity. An alternative way to conceptualize the mapping between a brain area and behaviour is illustrated in FIG. 4. A given brain area, A, is involved in multiple neural computations, NC. Note that this initial mapping is itself many-to-many, so that a given area (for example, A1) is involved in the computation of several functions (for example, NC1 and NC3), and a given computation (for example, NC3) might be implemented by several areas (for example, A1, A2 and A3). These neural com-putations collectively underlie behaviour. One can describe the space of behaviours using affective and cognitive axes. Thus, any behaviour is by definition both cogni-tive and affective. Importantly, the axes are not orthogonal, such that a behaviour that is changed along the affective dimension compared to a different behaviour, will also be changed along the cognitive dimension. In other words, behaviour cannot be cleanly separated into cognitive or emotional categories.

An important aspect of the present proposal is that individual brain areas do not work in isolation, but instead are part of net-works. Therefore, most neural computations should not be thought of as implemented by an individual area, but rather by the interac-tion of multiple areas. In addition, as pointed out by Mesulam126, specific brain areas belong to several intersecting networks. Thus, the computations implemented by an area will depend on the particular network with which the area is affiliated at a par-ticular time. For instance, areas A1 and A2 participate in the computation of NC3 when A1 and A2 are part of network N1 (FIG. 4). In another context, A2 might be affiliated with network N2, for example, whereas NC3 might be implemented in another manner. Thus, it is necessary to take into account the context in which neural computations are being carried out, because it depends on this context which brain areas will implement the computations.

To understand the impact of a region on behaviour, its connectivity pattern needs to be determined134. Recent advances in network theory135,136 have shown that regions characterized by a high degree of connectiv-ity (hubs137) are important in regulating the flow and integration of information between regions (area A2 in FIG. 4). However, whereas the number of connections of a region is important in determining whether it will function as a hub, the structural topology of a hub is also relevant. For instance, some regions are best characterized as ‘provincial’ hubs (they occupy a central position within a

single functional cluster, for example, visual area V4137) or as ‘connector’ hubs (they link separate region clusters, for example, frontal eye fields137).

In the present context, connectivity information could be particularly important in understanding a region’s role in emo-tion and cognition. Regions that are more peripheral (for example, non-hub regions with fewer connections) will often be described as either cognitive or emotional. This classification could be applied to provincial hubs, too. In both cases, regions whose function involves homeostatic proc-esses and/or bodily representations will be viewed as emotional, whereas regions whose

Figure 4 | Conceptual proposal for the rela-tionship between anatomical sites, neural computations and behaviours. Brain areas (for example, A2), which are connected to form net-works (ellipses), are involved in multiple neural computations (for example, NC2, NC3 and NC4) and specific computations (for example, NC4) are carried out by several areas (for example, A2 and A3). Therefore, the structure–function mapping is both one-to-many and many-to-one; in other words, many-to-many. Multiple neural computa-tions underlie behaviour. Each behaviour has both affective and cognitive components, indi-cated by the affective and cognitive axes. Note that the axes are not orthogonal, indicating that the dimensions are not independent from each other. Brain areas with a high degree of connec-tivity are called hubs and are critical for regulat-ing the flow and integration of information between regions. The structural topology of the hubs is strongly linked to their function. For instance, connector hubs (such as A2) link sepa-rate region clusters (not shown) and are hypoth-esized to be crucial for the integration of cogni-tion and emotion. See REF. 126 for a related scheme.

P E R S P E C T I V E S

154 | FEBRUARY 2008 | VOLUME 9 www.nature.com/reviews/neuro


Teorie e modelli: //processo di depolarizzazione della membrana

Teorie e modelli: //membrana: modello di Hodgkin-Huxley

Teorie e modelli: //membrana: modello di Hodgkin-Huxley

Teorie e modelli: //membrana: modello di Fitzhugh-Nagumo

Teorie e modelli: //il neurone: un modello semplificato

Il livello neurobiologico: neuroni //un modello semplificato

d e n d r i t e : r i c e v e i n f o r m a z i o n e d a i neuroni e la trasmette al corpo cellulare

soma o corpo cellulare: contiene il nucleo, con i cromosomi. Se il soma è sufficientemente eccitato dall’informazione proveniente dai dendriti trasmette un segnale all’assone

assone: trasmette l’informazione dal soma alle sinapsi



assone: trasmette l’informazione dal soma alle sinapsi





assone: trasmette l ’ in formazione dal soma alle sinapsi


Il livello neurobiologico: neuroni //la questione dei livelli

F-N

H-H





Modello teorico

Simulazione Livello: membrana





Modello teorico

Simulazione Livello: unità neurale





Modello teorico

Simulazione

dati sperimentali





Modello teorico

Simulazione

dati sperimentali

Output: spike





Modello teorico

Simulazione


Simulazione

Campionamentostocastico dell’input

“Pesatura sinaptica” e somma

Funzione a sogliaOutput: spike


Probability and Random Processes

A random variable is a number assigned to every outcome of an experiment. This could be theoutcome of the roll of a die, or the number of action potentials generated by a visual neuron duringa 1 sec stimulus presentation. The probability of getting each possible outcome is characterizedby a probability density function. For a fair die, there is a 1/6 probability of getting each possi-ble outcome. The familiar bell-shaped curve of the normal distribution is another example of aprobability density function.

The integral of a probability density function is called the cumulative distribution function.Cumulative distributions characterize the probability of getting an outcome less than or equal tosome specified value. For example, there is a 1/2 probability of getting a roll less than or equal to3 on a fair die.

A random process is a rule for assigning a function to every outcome of an experiment.For example, the voltage trace recorded from an intracellular electrode during a 1 sec stimuluspresentation might be considered a random process.

Instantaneous Firing Rate

Define , the neural response function, to be a bunch of impulses, one for each action potential:

where is the total number of spikes in the spike train, and are the times that each spike occurred.The unit impulse signal is defined as:

such that the integral of is one:

We would like to think of the neural response function as a random process. The neural responsefunction is completely equivalent to a list of the spike times in the spike train. Nevertheless, it isuseful for re-expressing sums over spikes as integrals over time. For example, we can write thespike count, the number of spikes fired between times and as the integral:

because each spike contributes to the integral.

The instantaneous firing rate (e.g., of a sensory neuron) can now be formally defined to be theexpectation of the neural response function, averaged over an infinite number of repeats (e.g., of

2

k = numero di spikes












2

funzione di risposta neurale

spike = impulso












2

numero di spikes trat1 e t2

the same stimulus presentation):

In practice, of course, you can not run an infinite number of trials. The function you get byaveraging over a finite number of trials, is an estimate of the instantaneous firing rate:

where is the number of trials and is the neural response function for each trial. This, ofcourse, is not a continuous function because it is just a sum of functions. You get a smoothfunction only in the formal limit with an infinite number of trials. Typically, when working withreal data, you would blur to make it smooth. We do not have to worry about that in this classbecause the theoretical/computational neuroscientist has the luxury of being able to just make upa continuous function, .

The average spike count can then be defined from the instantaneous firing rate:

(1)

This is equivalent, of course, to counting the spikes in each of a very large (i.e., infinite) numberof repeated trials, and then averaging those spike counts across the trials.

For sufficiently small intervals, when and , the average spike countcan be approximated by . Furthermore, can be reduced until the probability thatmore than one spike could appear in this interval is small enough to be ignored. In this case, theaverage spike count is equal to the probability of firing a single spike. That is, the probability of aspike occurring during a given brief time interval is equal to the value of the instantaneous firingrate during that time interval times the length of the interval:

(2)

Unlike the neural response function which provides a complete description of the neural re-sponse, the instantaneous firing rate is a highly reduced description. It is constructed by averagingthe neural response function over many repeated trials, to identify the “systematic component” ofthe response that is common to all trials. Other averages of the neural response function could beconstructed, for example, the response correlation function . The question is whether ornot it is worth the effort to keep track of anything other than the instantaneous firing rate.

Poisson Processes

Poisson processes are important in a variety of problems involving rare, random events in time orspace, e.g., radioactive emissions, traffic accidents, and action potentials.

3

firing rate





(1)



(2)


Poisson Processes


3

stima del firing rate





(1)



(2)


Poisson Processes


3


k = numero di spikes in (0,T)

numero di spikes trat1 e t2





(1)



(2)


Poisson Processes


3





(1)



(2)


Poisson Processes


3





(1)



(2)


Poisson Processes


3

Homogeneous Poisson Process

We will begin by assuming that the underlying instantaneous firing rate is constant over time.This is called a homogeneous Poisson process. Later we will treat the inhomogeneous case inwhich varies over time. Imagine that we are given a long interval and we place a singlespike in that interval at random. Then we pick a sub-interval of length . Theprobability that the spike occurred during the sub-interval equals .

Now let’s place spikes in the interval and find the probability that of them fall in thesub-interval. The answer is given by the binomial formula:

where and . If you have never seen this binomial formula before, look inany undergraduate level probability or statistics book. The binomial formula is what you use tocalculate the probability of events of a certain type out of trials, for example, the probability ofgetting 10 sixes out of 100 rolls of a fair die.

Next we increase and keeping the ratio constant. Since is the total numberof spikes and is the total time, is the mean firing rate, the average number of spikesper second. It can be shown that as , the probability that spikes will be in an interval oflength equals:

(3)

This is the formula for the Poisson probability density function. Given the mean firing rate ,the formula tells you the probability of having spikes during a time interval of length . Theformula is only correct when the spikes are completely independent of one another, i.e., that theyare placed randomly throughout the full time interval.

The spike count for a homogeneous Poisson process, dropping the time-dependence from Eq. 1,is given by:

(4)

for any interval of length . As expected, the average spike count equals the meanfiring rate times the duration. The variance of the spike count is a bit harder to derive but it turnsout that the result is the same, i.e.,

The ratio of the variance to the mean spike count is called the Fano factor,

(5)

The Fano factor characterizes the variability in the spike count. The fact that the spike count meanand variance are equal is a distinguishing characteristic of a Poisson process.

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(3)



(4)



(5)


4






(3)



(4)



(5)


4






(3)



(4)



(5)


4






(3)



(4)



(5)


4


0 20 40 60 80 100 1200

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Prob

abili

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)

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In ters p ike in terva l (m s ec)

Figure 1: A. Snippet of a Poisson spike train with and msec. B. Spike counthistogram calculated from many Poisson spike trains, each of 1 sec duration with , superim-posed with the theoretical (Poisson) spike count density. C. Interspike interval histogram calculatedfrom the simulated Poisson spike trains superimposed with the theoretical (exponential) interspikeinterval density. D. Snippet bursty spike train generated by replacing each spike in A with a “burst”of zero, one, or more spikes. The height of each impulse represents the number of spikes in that timebin. The number of spikes per burst was Poisson distributed with a mean of spike/burst. E.Spike count histogram calculated from many bursty spike trains like that in D, superimposed withthe Poisson spike count density. The bursty spike trains have the same mean spike count, but thevariance of the bursty spike count histogram is twice that of the Poisson. F. Renewal process spiketrain generated from A by removing all but every fourth spike. G. Spike count histogram calculatedfrom many renewal spike trains like that in F. The mean spike count is 1/4 that of the Poisson, asexpected. H. Interspike interval histogram calculated from the renewal process spike trains super-imposed with the theoretical (gamma) interspike interval density. The mean interspike interval is40 msec, four times longer than that in C, as expected because we have removed 3/4 of the spikes.The standard deviation of the interspike intervals is 20 msec so the coefficient of variation is 1/2.

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0 20 40 60 80 100 1200

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In ters p ike in terva l (m s ec)

Figure 1: A. Snippet of a Poisson spike train with and msec. B. Spike counthistogram calculated from many Poisson spike trains, each of 1 sec duration with , superim-posed with the theoretical (Poisson) spike count density. C. Interspike interval histogram calculatedfrom the simulated Poisson spike trains superimposed with the theoretical (exponential) interspikeinterval density. D. Snippet bursty spike train generated by replacing each spike in A with a “burst”of zero, one, or more spikes. The height of each impulse represents the number of spikes in that timebin. The number of spikes per burst was Poisson distributed with a mean of spike/burst. E.Spike count histogram calculated from many bursty spike trains like that in D, superimposed withthe Poisson spike count density. The bursty spike trains have the same mean spike count, but thevariance of the bursty spike count histogram is twice that of the Poisson. F. Renewal process spiketrain generated from A by removing all but every fourth spike. G. Spike count histogram calculatedfrom many renewal spike trains like that in F. The mean spike count is 1/4 that of the Poisson, asexpected. H. Interspike interval histogram calculated from the renewal process spike trains super-imposed with the theoretical (gamma) interspike interval density. The mean interspike interval is40 msec, four times longer than that in C, as expected because we have removed 3/4 of the spikes.The standard deviation of the interspike intervals is 20 msec so the coefficient of variation is 1/2.

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