Introduction to Mathematical Methods in Neurobiology: Dynamical Systems Oren Shriki 2009 Modeling Conductance-Based Networks by Rate Models 1

References: Shriki, Hansel, Sompolinsky, Neural Computation 15, 1809 – 1841 (2003) Tuckwell, HC. Introduction to Theoretical Neurobiology, I&II, Cambridge UP,

Conductance-Based Models vs. Simplified Models There are two main classes of theoretical approaches to the behavior of neural systems: –Simulations of detailed biophysical models. –Analytical (and numerical) solutions of simplified models (e.g. Hopfield models, rate models). Simplified models are extremely useful for studying the collective behavior of large neuronal networks. However, it is not always clear when they provide a relevant description of the biological system, and what meaning can be assigned to the quantities and parameters used in them. 3

Conductance-Based Models vs. Simplified Models Using mean-field theory we can describe the dynamics of a conductance-based network in terms of firing-rates rather than voltages. The analysis will lead to a biophysical interpretation of the parameters that appear in classical rate models. The analysis will be divided into two parts: –A. Steady-state analysis (constant firing rates) –B. Firing-rate dynamics 4

Network Architecture External Inputs (independent Poisson processes) Recurrent connectivity 1234N f 1 inp f1f1 f N inp fNfN 5

Voltage Dynamics for A Network of Conductance-Based Point Neurons We assume that the neurons are point neurons obeying Hodgkin-Huxley type dynamics: I active – Ionic current involved in the action potential I ext – External synaptic inputs I net – Synaptic inputs from within the network I app – External current applied by the experimentalist 6

t t_spike A spike at time t spike of the presynaptic cell contributes to the postsynaptic cell a time-depndent conductance, g s (t): Synaptic Conductances Peak conductance 7

Synaptic Dynamics: An Example Synaptic dynamics are usually characterized by fast rise and slow decay. The simplest model assumes instantaneous rise and exponential decay: (Presynaptic rate) 8

Synaptic Dynamics: An Example For a single presynaptic spike the solution is: 9

Synaptic Dynamics: An Example Implementation in numerical simulations: Given the time step dt define the attenuation factor: A dimensionless parameter, f, is increased by 1 after each presynaptic spike and multiplied by the attenuation factor in each time step. The conductance is the product of f and the peak conductance, G. 10

Synaptic Dynamics For simplicity, we shall write in general: K(t) is the time course (dimensionless) function. We define: For example: 11

External Synaptic Current The explicit expression for the external synaptic current is: The peak synaptic conductance is: The time constant is: 12

Internal Synaptic Current The explicit expression for the internal synaptic current is: The peak synaptic conductance is: The time constant is: 13

Part A: Steady-State Analysis The main assumptions are: Firing rates of external inputs are constant in time Firing rates within the network are constant in time The network state is asynchronous The network contains many neurons 14

Asynchronous States in Large Networks In large asynchronous networks each neuron is bombarded by many synaptic inputs at any moment. The fluctuations in the total input synaptic conductance around the mean are relatively small. Thus, the total synaptic conductance in the input to each neuron is (approximately) constant. 15

Asynchronous States in Large Networks The figure below shows the synaptic conductance of a certain postsynaptic neuron in a simulation of two interacting populations (excitatory and inhibitory): 16

Mean-Field Approximation We can substitute the total synaptic conductance by its mean value. This approximation is called the “Mean-Field” (MF) Approximation. The justification for the MF approximation is the central limit theorem. 17

The Central Limit Theorem A random variable,which is the sum of many independent random variables, has a Gaussian distribution with mean value equal to the sum of the mean values of the individual random variables. The ratio between the standard deviation and the mean of the sum satisfies: where N is the number of individual random variables. 18

Mean-Field Analysis of the Synaptic Inputs The total contribution to the i’th neuron from within the network is: The contribution to this sum from the j’th neuron is: 19

Mean-Field Analysis of the Synaptic Inputs Consider a time window T>>1/f j, where f j is the firing rate of the presynaptic neuron j. Schematically, the contribution of the j’th neuron in this time window looks like this: spikes 20

Mean-Field Analysis of the Synaptic Inputs The mean conductance due to the j’th neuron over a long time window is: The mean conductance resulting from all neurons in the network is: 21

Mean-Field Analysis of the Synaptic Inputs The mean-field approximation is: Effectively, we replace a spatial averaging by a temporal averaging. Using a similar analysis, the contribution of the external inputs can be replaced by: 22

Mean-Field Analysis of the Synaptic Inputs The synaptic currents are not constant in time since the voltage of the neuron varies significantly over time: We can decompose the last expression in the following way: 23

Mean-Field Analysis of the Synaptic Inputs We now use the MF approximation : This gives: A constant applied current A constant increase in the leak conductance 24

Mean-Field Analysis of the Synaptic Inputs Similarly: We obtained the following mapping: 25

Mean-Field Analysis of the Synaptic Inputs To sum up: Asynchronous Synaptic Inputs Produce a Stationary Shift in the Voltage-Independent Current and in the Input Passive Conductance of the Postsynaptic Cell. To complete the loop and determine the network’s firing rates we need to know how the firing rate of a single cell is affected by these shifts. 26

Current-Frequency Response Curves of Cortical Neurons are Semi-Linear Excitatory Neuron (After: Ahmed et. al., Cerebral Cortex 8, , 1998): Inhibitory Neurons (After: Azouz et. al., Cerebral Cortex 7, , 1997) : 27

The Effect of Changing the Input Conductance is Subtractive Experiment: f-I curves of a cortical neuron before and after iontophoresis of baclofen, which opens synaptic conductances. (Connors B. et. al., Progress in Brain Research, Vol. 90, 1992). 28

A Hodgkin-Huxley Neuron with an A-current 29

The Addition of a Slow Hyperpolarizing Current Produces a Linearization of the f-I Curve ____ - No A-current (g A =0) ____ - Instantaneous A-current (g A =20,  A =0 ) ____ - Slow A-current (gA=20,  A =20 ) [gA]=mS/cm2, [  A]=msec g A =20,  A =20, I=1.6 [  A/cm 2 ] 30

The Effect of Increasing gL is Subtractive 31

Model Equations with Parameters: Shriki et al., Neural Computation 15, 1809–1841 (2003) 32

The Dependence of the Firing Rate on I and on g L Can Be Described by a Simple Phenomenological Model [x] + =x if x>0 and 0 otherwise. We find for the model neuron :  =35.4 [Hz/(  A/cm2)] V c =5.6 [mV] I C 0 =0.65 [  A/cm2] 33

The Effect of the Synaptic Input On the Firing Rate Combining the previous results, we find that the steady-state firing rates obey the following equations: 34

The Effect of the Synaptic Input On the Firing Rate This can be written us: Where: 35

The Units of the Interactions The units of J are units of electric charge: The quantity J ij f j reflects the mean current due to the j’th synaptic source. The strength of the interaction J ij reflects the amount of charge that is transferred with each presynaptic action potential. 36

‘ excitatory ’ ‘ inhibitory ’ The Sign of the Interaction The interaction strength in the rate model has the form: The rule for excitation / inhibition is: This does not necessarily coincide with the biological definition of excitation/inhibition. 37

The Sign of the Interaction The biological definition is: A positive J implies that this synaptic source increases the firing rate. In general, it may be that a certain synaptic source tends to elicit a spike but increases the conductance in a way that reduces the overall firing rate. excitatory inhibitory 38

The Model Parameters Neuron: β – Slope of frequency-current response V c, I c 0 – Dependence of current threshold on leak conductance E L – Reversal potential of leak conductance Synapse: G ij – Peak synaptic conductance E j – Synaptic reversal potential τ ij – Synaptic time constant 39

Rate Model for a Homogeneous, Highly Connected Excitatory Network: 40

Simulations of the Excitatory Network and Rate Model Prediction 41

A Model of a Hypercolumn in the Primary Visual Cortex 42

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Neurons in the Retina and in the Thalamus are Sensitive to Circular Spots of Light 45

Neurons in the Primary Visual Cortex are Sensitive to Oriented Stimuli 46

Orientation [deg] Cell Activity Neurons in the Primary Visual Cortex are Sensitive to Oriented Stimuli Contrast levels 47

Orientation Map – A Scheme Hypercolumn 48

Orientation Map – Experimental Results 49

What is the Mechanism Behind Orientation Tuning? Thalamus Cortex A Feed-forward model (Hubel & Wiesel)

Horizontal Connections in V1 51

A Model of a Hypercolumn in the Primary Visual Cortex 52

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Intermediate Summary We constructed rate models for conductance-based networks under two main assumptions: a) The network is large and asynchronous. b) The f-I curve shifts when the input conductance is changed - but does not change shape. Take Home Message: The steady-state response properties of cortical networks may be well predicted by analytically amenable simple rate models. Question: How to extend this approach to study non-stationary behavior of large neuronal systems? 57

Rate Dynamics Two dynamical processes: 1.The dynamics of the firing rates given the synaptic activity 2.The dynamics of the synaptic activity given the firing rates 58

Dynamics of synaptic Conductances We define a normalized synaptic activity by: 59

Dynamics of synaptic Conductances The synapse acts as a low-pass filter. 60

A Naive Model for the Firing Rate Dynamics 61

A Naive Model for the Firing Rate Dynamics It naïve model predicts that for sinusoidal inputs the amplitude and phase of the response will not depend on the modulation frequency. Simulations show that they do depend  the naïve model fails! 62

Single Neuron Responses to Sinusoidal Inputs 63

Dependence of Resonance Frequency on Mean Firing Rate 64

A Model for the Firing Rate Dynamics It is instructive to replace the instantaneous current and conductance by filtered versions. Numerical simulations show that a band-pass filter (a second order linear filter) gives a good approximation. 65

Response to a Broadband Input 66

Excitatory Network 67

Inhibitory Network 68