Network Effects of Synaptic Modifications

Abstract

In this paper, we use computational models of varying complexity to investigate the role of synaptic modifications for cortical network properties. In particular, we study how the dynamics can be regulated by neuromodulators, intrinsic noise and chemical agents. We focus on the complex neurodynamics and its modulation, and how this is related to the neural circuitry, where connectivity enhancement and pruning is considered. The emphasis is on the overall network structures, with feedforward and feedback loops between excitatory and inhibitory neurons at several layers and distances, and less details at the synaptic level. Our models aim at linking processes at a molecular and cellular (microscale), with processes at a network level (mesoscale), which in turn are linked to the mental processes and cognitive functions (macroscale). We also discuss the relevance of these results for clinical and experimental neuroscience, with applications to learning, memory, arousal, and mental disorders.

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Appendix A

Equations for the paleocortical network model with population units

The time evolution for a network of N neural units is given by a set of coupled nonlinear first order differential delay equations for all the N internal states, u. With external input, I(t), characteristic time constant, τ_i , and connection weight w _ij between units i and j, separated with a time delay δ _ij , we have for each unit activity, u _i ,

The input-output function, g _i (u _i ), is a continuous sigmoid function, experimentally determined by Freeman [18]:

The gain parameter Q _i determines the slope, threshold and amplitude of the curve for unit i. This gain parameter is associated with the level of arousal, or alternatively, the level of any particular neuromodulator. C is a normalization constant. Neuronal adaptation is implemented as an exponential decay of the output, proportional to the time average of previous output [50]. In such cases, C is not a constant, but instead denotes the adaptation function, and the input–output relation becomes,

where <> _T denotes the time average over the last T ms, and α is an adaptation parameter that is under neuromodulatory (cholinergic) control. Noise, or spontaneous neural activity is represented by a Gaussian noise function, ξ _t (t), such that <ξ _t (t)≥and <ξ _t (t)ξ _t (s)≥2Aδ(t–s).

The connection weights w _ij are initially set and constrained by the general connectivity principles that have evolved for the three-layered cortex. To allow for learning and associative memory, the connection weights, w _ij are incrementally changed, according to a learning rule of Hebbian type [34], adapted for the system dynamics. It takes into account that there is a conduction delay, δ _ij , between the output (presynaptic) activity of one network unit and its (postsynaptic) effect on the receiving unit. With learning rate η the change in connection weight between unit j and i is given by

where w_max is the maximum strength of an intrinsic synaptic connection.

Appendix B

Equations for the network model with Frankenhaeuser-Huxley neurons

The model neuron is described by the Frankenhaeuser-Huxley equations, with parameters based on recordings from small hippocampal interneurons [41]. The FH-model deviates slightly from the classical Hodgkin-Huxley formalism, since the FH-equations are based on recordings from vertebrate axons, whereas the HH-equations are based on recordings from squid axons. The FH-model is based on four differential equations:

where v is the membrane potential, and m, h and n are permeability variables for P _Na activation (m), P _Na inactivation (h) and P _K activation (n). The variation in the membrane potential is due to the stimulation current (I _S ), the leak current (I _L ), the initial transient current through Na channels (I _Na ) and the delayed sustained current through K channels (I _K ) divided by the membrane capacitance (C _M ). The α's and, β's are rate constants for m, h, and n, as indicated by the suffices. A _m is the membrane area, R the gas constant, F is Faraday's constant and T the absolute temperature. [Na] _o and [Na] _i are Na concentrations. Parameter values and the equations for I _Na , I _K , I _L are given in [41] and in [6]. In our simulations, the only free parameters for the neuronal model are P* _Na and P* _K , the permeability values for fully open ion channels, also referred to as the channel densities. For simplicity, we skip the dimensions when referring to the permeability constants (or channel densities), always giving the values in units of 10⁻⁶ m/s.

Our network model consists of 6×6 neurons, arranged in a square lattice and connected in an all-to-all manner. We use a distance-dependent connectivity, with the connection strength decreasing with distance as w∼1/r (although in reality such effects may be negligible at this scale). In the cases when inhibition is included, we let six (out of 36) homogenously distributed network neurons be inhibitory (with periodical boundary conditions), relating to the fact that about 20% of the neocortical neurons in the mammalian brain are inhibitory.

We have previously studied the dynamical effects of changing the balance between excitation and inhibition in cortical neural networks [25]. In the current work, we also study the effects that different neuronal types (in terms of channel densities) could have on the network activity, by letting excitatory and inhibitory neurons having different properties. For simplicity, we let the post-synaptic activity be the same for inhibition as for excitation, except for opposite signs.

There is a synaptic delay of 1 ms in the signal transfer, while delays due to axonal propagation are neglected for the short distances considered here. The signal transfer is modelled in an all-or-nothing manner. A neuron is considered to elicit an action potential when the sodium activation exceeds a certain threshold (m>0.6). The effect of an action potential is a stereotyped postsynaptic current that is weighted and given as input to the receiving neurons. For the postsynaptic current, we follow the approach of Gerstner (2000), using an exponential pulse. A single action potential (f) elicited by neuron j thus results in a postsynaptic input to neuron i, as given by the equations:

where I ^(f) _ij (t) is the input to neuron i from neuron j at time t, as resulting from the action potential at time t ^(f) . t _syn (1 ms) is the synaptic delay, and τ _syn is the synaptic (membrane) time constant (30 ms). The connection weight (w _ij ) for the signal transfer to neuron i from neuron j decreases proportionally with the distance (d _ij ) between the neurons (normalized by use of the nearest neighbour distance d ₀ =0.2 μm). w _ij is a measure of the efficacy of the synapse between the neurons i and j, and has the unit of charge. Since the remaining factor of equation (6) is normalized, w _ij can be identified with the charge deposited on the capacitor by a single presynaptic pulse from neuron j. q (=2.3*10⁻¹⁰ C) is just a scaling constant, chosen in order to have w=1 as a unitless reference. The weight factor, w, is a key parameter in our simulations, since it is regarded as a global measure of connection strength. At w=1, the initial postsynaptic current is about 7.75 pA, which is also a typical value for the stimulation current. Since the synaptic input is decaying with distance, it normally requires several AP inputs in order to generate an AP in the receiving neuron.

When a presynaptic neuron oscillates, the postsynaptic contributions from preceding action potentials will simply add up in the inputs. The total synaptic input to a neuron i will be the sum over action potentials (f) of all the neurons (j) that give an input to neuron i:

The synaptic input will enter the single neural model as an additional input current. Equation (1) thus becomes:

In the network simulations, we also include a white noise term (I _G ), using a sampling rate of 0.1 ms and drawing the noise amplitude from a Gaussian distribution, with mean 0 and a variance equal to 5pA.

Appendix C

Equations for the neocortical network model with FitzHugh-Nagumo units

In our models of ECT EEG, the network dynamics are described by Eq. (1). The neurons are modelled as continuous output units of a Fitzhugh-Nagumo type, described by Eq. (2) and Eq. (3), which are essentially the same as in Ref. [4].

In Equations (1–3), n is the number of neurons, ui is the postsynaptic potential of neuron i, vi is the membrane potential at the axon initial segment, and wi is an auxiliary variable. a, b, and c are positive constants, which guarantee the existence of an oscillatory interval.

is a monotonically increasing (α>0), nonnegative (0≤m_g<M_g ) function that describes the relation between the pre- and postsynaptic potentials of the neurons. c_ik describes the topology of the network. p ⁺ and p ⁻ stand for the excitatory and inhibitory connection strengths. The neurons have time-constants τ ^ex and τ ⁱⁿ . The total time delay, T _ik , is due to a synaptic delay, Ts=1 ms, and the time a signal takes to travel through the axonal and dendritic trees between neuron k to neuron i with the propagation velocity 0.5 m/s. γ _i denotes the synaptic membrane conductance of neuron i. We take the mean membrane potential of all excitatory neurons in each layer in the system as a model of the EEG readout.

Correspondence