Computational cognitive neuroscience: 2. Neuron. Lubica Beňušková Centre for Cognitive Science, FMFI Comenius University in Bratislava

Transcription

1 1 Computational cognitive neuroscience: 2. Neuron Lubica Beňušková Centre for Cognitive Science, FMFI Comenius University in Bratislava

2 2 Neurons communicate via electric signals In neurons it is important where the membrane potential crosses a certain threshold. That is considered an important event for the transmission of the signal from one cell to another.

3 Presynaptic and postsynaptic: flow of information soma axon dendritic tree spines basal dendrites synapse Axons of presynaptic neurons make synapses (little blobs) on the body and dendrites of the postsynaptic neuron 3 3

4 Neurons in the networks communicate via synapses Presynaptic electric pulse (i.e. spike) arrives to terminal. This causes release of neurotransmitter molecules from vesicles. Neurotransmitter binds to receptors in the dendritic membrane. Ion channels associated with receptors open and allow influx of charged ions into the spine evoking a postsynaptic potential at synapse. axon of presynaptic neuron dendrite of postsynaptic neuron 4

5 5 Inhibitory synapses Neutrotransmitter GABA, 2 types of postsynapstic receptors GABRA and GABRB, ion channels for Cl - and K +, respectively. Ca 2+ Presynaptic terminal GABA Cl - K + When we measure an electric potential in the postsynaptic site, we see something like this: a negative deviation from the resting potential, which is called inhibitory postsynaptic potential (IPSP). GABR A R N GABR B

6 6 Excitatory synapses Ca 2+ Presynaptic terminal R Glutamate Na + Na + AMPA N NMDA Ca 2 + Neutrotransmitter glutamate, 2 types of postsynaptic receptors AMPA and NMDA, ion channels for Na + and Na + and Ca 2+, respectively. When we measure an electric potential in the postsynaptic site, we see something like this: a positive deviation from the resting potential, which is called excitatory postsynaptic potential (EPSP).

7 7 Summation of individual PSPs Each presynaptic electric pulse (spike) causes a single EPSP bump at the postsynaptic site. On arrival of many presynaptic spikes in close succession, these EPSPs sum on top of each other and if the cumulative sum crosses the firing threshold Q, the neuron generates an output spike (also called an action potential). (Q) EPSP S EPSP S EPSP

8 Neuron: an analog-digital converter Q EPSP IPSP EPSPIPSP > Q Since the inhibitory synapses can be activated as well, the total V at soma = SEPSP-SIPSP. If PSP > Q, neuron generates an output spike train during the time when the total PSP at soma > Q. We call it a spike train. The number and frequency/rate of spikes within the output spike train are proportional to the duration and magnitude of V at the soma. Spike train then propagates towards synapses, where it causes release of neurotransmitter. 8

9 Simultaneous recordings of output spike trains of 30 neurons from monkey visual cortex [Krüger and Aiple, 1988]. Postsynaptic activity of real neurons Each spike is denoted by a vertical bar, with a separate row for each neuron. An interval of 150 msec is shaded. This time span is known to suffice for the completion of some very complex computations. Neurons use few spikes to accomplish complex tasks. 9

10 10 Ultrafast visual classification of objects Simon Thorpe et al. at the University of Toulouse, France, performed an experiment with humans and monkeys, in which subjects were supposed to classify pictures into 2 categories, either an animal category or a non-animal category (Thorpe et al., Science, 1996). Hundreds of pictures were used. Subjects did not see the pictures before. Every 2 s, a picture from one or the other class was drawn at random. Each time, it was a different picture, thus there was no repetition. Pictures were shown just for 20 ms. In spite of that, humans on average correctly classified the pictures in 94% cases and monkeys in 91%. Ultra-fast classification did not depend on classes of objects, did not depend on colour, and did not depend on attention or eye fixation.

11 11 Ultrafast visual classification of objects During an experiment the brain activity of subjects was recorded. Reaction time, i.e. time from presentation of stimulus to pressing the button, was equal to 250 ms in humans, on average. Prefrontal cortex Frontal cortex PMC 250 ms MC Higher-order parietal visual areas Activity in the inferotemporal (IT) cortex occurred on average after 150 ms, so the preparation and execution of motor response took on average 100 ms. Eye Thalamus IT 150 ms V4 V2 V1 Cerebellum Primary visual cortex In monkeys, times were shorter by about 80 ms (smaller brains).

12 12 Ultrafast visual classification of objects Projected image stimulates retina for 20 ms. In about 80 ms, thalamic LGN responds. Thalamic neurons activate neurons in the primary visual cortex (V1). Then, activation proceeds to and through higherorder visual areas, V2, V4 and IT, where activity appears after 150 ms. Thus (150-80) / 4 areas = 17.5 ms per area. Even if neurons fire with f = 100 Hz, this would mean that each one fires 1-2 spikes within this interval Prefrontal cortex Eye Frontal cortex PMC 250 ms MC Thalamus IT 150 ms V4 Cerebellum Higher-order parietal visual areas V2 V1 Conclusion: individual neurons do not have time to calculate mean rates of firing!!! Primary visual cortex

13 13 Rate or frequency code These models are based on the idea that a feature of an object is coded by the frequency of spikes. In the figure, each of the five neurons codes a different feature of an object. Neuron will only reach the threshold needed to generate an output when the total number of spikes it receives over some time exceeds some value.

14 14 Synchrony code (coincidence detection) Features that are represented by particular populations of neurons will synchronize the firing of these neurons if the features belong to the same object (binding by synchrony). In the figure, each of the 5 neurons codes for a different feature of the same object. Neurons act as coincidence detectors thus responding to and strengthening inputs that are active at the same time.

15 15 Spike delay code Delay coding is based on the idea that neurons will respond to more energetic input signals by generating a spike earlier, that this generated spike will arrive at the downstream neuron earlier than other spikes and will thus have a higher impact or 'ranking' relative to later incoming spikes as a result. Another variant is that information is encoded in the phase relationship between the cell s firing activity and the underlying rhythmic oscillations of the whole population of neurons.

16 16 Spatio-temporal coding Neurons respond to the same stimulus with the same stimulus-specific spatio-temporal pattern of spikes. Different stimuli evoke different spatio-temporal patterns of spikes in neuronal populations that process them.

17 17 Neural code hypotheses a. Rate. Information about the feature is encoded in the frequency of neuron s firing (Adrian 1929, Hubel and Wiesel 1962). b. Synchronisation. Populations of neurons that represent features belonging to one object are bound together by synchronous firing (Singer et al. 1996). Synfire chains: information is carried by waves of synchronously firing neural ensembles (Bienenstock 1995, Abeles 2001). c. Delay code. Phase. Information about the feature is encoded in the phase of neuron s impulses with respect to the reference background oscillation (Jensen 2001, Hopfield 2005). Time to the first spike. Neurons that fire the first carry the information about the stimulus features. The rest of neurons and the rest of impulses are ignored (Thorpe et al. 1996). d. Spatio-temporal pattern of firing. Information about the salience of the object feature is encoded in the exact temporal structure of the spike train (Rieke et al. 1996, A. Villa 2005).

18 Spiking versus rate model of neurons Spiking Neuron Rate model of neuron PSP I 1 Out time I 2 Sigmoid O 1 D 1 D 2 D 3 S Th () I 3 Input Output Input Output Spiking neuron receives spikes with various delays D j Input spikes induce either EPSP or IPSP, which add upon each other Output: a single spike or a train of spikes when SEPSP-SIPSP > Q Inputs to the rate model of a neuron are numbers (e.g., 0.6, 0.7, etc.) that represent the frequency of incoming spike trains Output: the rate-based model produces a single number (e.g., 0.4) that represents the rate of the output spike train. 18

19 19 How do we construct a spiking model of a neuron? Important note: in all types of model neurons there is an important variable, which we call the weight of a synapse that reflects the strength of input coming from the presynaptic neuron j upon the postsynaptic neuron i.

20 20 Integrate and fire (I&F) models Soma plus dendrites are modelled as a single compartment. The dependent variable is the membrane voltage V, the independent variable is time t. Action potentials occur when the membrane potential V reaches a firing threshold Q. Leaky I&F: after firing, the membrane potential is reset to the value of resting potential V 0. I&F can be modelled at various levels of rigour depending on simplifying assumptions used.

21 Single compartment model Electric scheme equivalent to an integrate and fire (I&F) neuron model C m is the capacitance of the membrane, g k is the conductance for ion of type k, and E is the equilibrium potential for a given type of an ion. Rate of change of the membrane potential V is proportional to the electric current entering into neuron; by convention capacitance current is plus while membrane current is minus : c dv dt m i m 21

22 22 Membrane current i m Membrane current i m is the sum of leakage current (L) and all the ion currents flowing through all the currently open ion channels per unit area of the membrane (thus k = Na +, K +, and Cl ): i m i L k i k i L I Each current component is equal to driving force (difference between the membrane potential V and the equilibrium potential E k ) multiplied by the corresponding ion channel conductance g k, where k stands for ions, i.e. k = Na +, K +, and Cl : i i k L g V E k ( k g L ( V EL ) )

23 23 Integrate & fire model of a spiking neuron The simplest model of a spiking neuron is described by this equation: c c m dv dt dv dt m i m g ( V E ) L ; if V Q a postsynaptic spike is fired L Where: I = S i k is the sum of all excitatory and inhibitory ionic currents I mediated by ions of type k; the symbol L denotes the so-called leak current that is always present, b/c the membrane always leaks some currents. Dependent variables are: V, ionic conductances g k, and I. Parameters are: c m, g L, Q, and all the equilibrium potentials E.

24 24 Adaptive exponential integrate-and-fire model AdEx O Reilly uses AdEx introduced by Gerstner and Brette: c m dv dt f ( V ) w dw dt f ( V ) g L ( V w E L I ) g a( V E ) w L L D T V exp Q D T if V Q, then three events happen Postsynaptic neuron fires a spike V is reset to a reset potential, for instance V 0 and w is reset to w + b Dependent variables are: V, I, and variable w. Parameters are: c m, g L, Q, D T, a, b, and all the equilibrium potentials E.

25 25 How do we fix the model parameter values? Step 1: Fix the known parameters (e.g. V 0, E L, etc) and make educated guesses for the remaining unknown parameter values. Step 2: Use the model to simulate experiments, producing model data. Step 3: Compare the model data with experimental data. Step 4: Adjust one or more parameter values and repeat from Step 2 until the simulated data sufficiently matches experimental data. Step 5: Use the model to simulate new experiments not used in the steps above and compare the resulting model data with the new experimental data to verify that the chosen parameter values are robust.

26 26 Behaviour of I&F model in response to excitatory inputs Firing threshold Q Spike waveforms are simply superimposed upon underlying waveform of membrane potential V Differential equation for V can be solved numerically for different time course of synaptic current I e as shown in the figure above. Here we only consider excitatory synaptic currents, hence the subscript e in I e

27 A big leap: Izhikevich s model of spiking neuron reproduces firing behavior of many types of cortical neurons. It combines the biologically plausibility of Hodgkin-Huxley-type dynamics and the computational efficiency of integrate-and-fire neurons. ( ) if dv dt du dt v 5v 140 u v a( bv u) peak, v c then u u d I Dependent variables: I, v and u Parameters: a, b, c, d these determine a particular spiking pattern Parameter peak is the value of the spike peak (see the next slide) 27

28 The project: Simple spiking model of Izhikevich 28

29 29 Reply of Izhikevich neuron to a constant input Let s image I = constant, then the regularly spiking cell will produce this output The so-called chattering neuron responds to I = constant like this And the low-threshold (usually an inhibitory) neuron responds like this

30 30 Reply of Izhikevich neuron to random input Let s imagine I = random, then the regularly spiking cell will produce this output But the so-called chattering type of a neuron responds like this And the low-threshold (usually an inhibitory) neuron responds like this

31 31 Applications The Gerstner and Brette s model is used in the Blue Brain project Izhikevich has developed his model based on a deep dynamical analysis of general properties of neurons, see Dynamical System in Neuroscience: E.I. is a director of the Brain Corporation: the goal of which is to build artificial brains for commercial applications.

32 32 Summary Since we ignore many details, including the geometry of the dendrites, and particular ion channels in the membrane, then we need to adjust the values of parameters in the right hand side of the basic equation for the voltage V. Thus, there are many modifications of the basic I&F model Adaptive exponential I&F model AdEx of Brette and Gerstner Simple spiking model of Izhikevich And many more