Part 5 · Chapter 5.5

Neural Integration

A cortical neuron receives ~10 000 synaptic inputs and decides, based on their integrated effect at the axon hillock, whether to fire an action potential. Integration involves spatial summation (simultaneous inputs from different locations), temporal summation (repeated inputs), excitatory-inhibitory balance, and dendritic signal processing that make single neurons nonlinear computing elements.

1. Spatial & Temporal Summation

The soma integrates EPSPs/IPSPs according to cable-theory attenuation and membrane leak. The axon hillock has the lowest threshold (densest Nav1.6) and fires an AP when integrated voltage crosses ~−55 mV:

\[ V_{soma}(t) = \sum_i w_i\,\text{EPSP}_i(t - t_i) + \sum_j w_j\,\text{IPSP}_j(t - t_j) \]

Weights w_i reflect dendritic attenuation (distal synapses heavily discounted) and synapse strength (determined by AMPA receptor number). Passive membrane time constant τ = R_mC_m (~10 ms) sets the temporal window for summation.

Simulation: Three Integration Modes

Python

script.py50 lines

import numpy as np, matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# Spatial summation: 3 EPSPs at different dendritic distances
t = np.linspace(0, 30, 500)
def epsp(t, t0, decay=5):
    return np.maximum(0, 4*np.exp(-(t-t0)/decay) - 4*np.exp(-(t-t0)/0.5)) * (t>t0)

# Spatial summation - simultaneous EPSPs
spatial = epsp(t, 2) + 0.7*epsp(t, 2) + 0.5*epsp(t, 2)

# Temporal summation - sequential EPSPs
temporal = epsp(t, 5) + epsp(t, 8) + epsp(t, 11) + epsp(t, 14)

# Excitation-inhibition
EPSP = epsp(t, 3)*2 + epsp(t, 7)*1.5 + epsp(t, 11)*2
IPSP = -0.8 * (epsp(t, 4) + epsp(t, 8))
EI = EPSP + IPSP

fig, axes = plt.subplots(1, 3, figsize=(16, 5), facecolor='#0a0f1e')
for ax in axes:
    ax.set_facecolor('#111e2f'); ax.tick_params(colors='#bae6fd')
    for s in ax.spines.values(): s.set_color('#334155')
    ax.grid(True, color='#334155', alpha=0.3)

axes[0].plot(t, spatial, color='#34d399', lw=2.5)
axes[0].axhline(15, color='#f87171', ls='--', label='Spike threshold')
axes[0].set_xlabel('Time (ms)', color='#bae6fd')
axes[0].set_ylabel('Somatic EPSP (mV)', color='#bae6fd')
axes[0].set_title('Spatial summation', color='#67e8f9', fontweight='bold')
axes[0].legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

axes[1].plot(t, temporal, color='#fbbf24', lw=2.5)
axes[1].axhline(15, color='#f87171', ls='--', label='Threshold')
axes[1].set_xlabel('Time (ms)', color='#bae6fd')
axes[1].set_title('Temporal summation', color='#67e8f9', fontweight='bold')
axes[1].legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

axes[2].plot(t, EPSP, color='#34d399', lw=2, label='EPSPs alone')
axes[2].plot(t, EI, color='#60a5fa', lw=2.5, label='EPSPs + IPSPs')
axes[2].axhline(15, color='#f87171', ls='--')
axes[2].set_xlabel('Time (ms)', color='#bae6fd')
axes[2].set_title('E-I balance', color='#67e8f9', fontweight='bold')
axes[2].legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

plt.tight_layout()
plt.savefig('output.png', dpi=120, bbox_inches='tight', facecolor='#0a0f1e')
print('Axon hillock integrates membrane voltage; fires if V crosses threshold')
print('Dendritic attenuation: EPSP from distal spine ~10% by soma')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

2. Dendritic Nonlinearities

Dendrites are not purely passive cables. NMDA-dependent spikes in spines, voltage- gated Na⁺ and Ca²⁺ channels in dendrites, and back- propagating action potentials produce local nonlinear computations. Individual dendritic branches can act as independent coincidence detectors (Poirazi 2003). A single pyramidal cell is a multi-layer computational unit, not a single threshold neuron.

3. Rate Coding vs Temporal Coding

Information in spike trains can be represented by firing rate (Adrian 1926) or by precise spike timing (Rieke 1997). Both coexist: motor cortex uses rate coding for movement direction; olfactory cortex uses precise temporal sequences; hippocampal place cells encode position through theta-phase precession. Decoding strategy depends on the downstream synapse.

4. Synaptic Plasticity

Long-term potentiation (LTP) and long-term depression (LTD) change synaptic weights on timescales of minutes to years. LTP requires NMDA-receptor Ca²⁺influx + postsynaptic depolarisation — a Hebbian coincidence detector. Spike-timing-dependent plasticity (STDP) makes weight change dependent on the order of pre- and postsynaptic spikes. This is the molecular substrate of learning and memory (Bliss & Collingridge 1993; Kandel Nobel 2000).

Key References

• Magee, J. C. (2000). “Dendritic integration of excitatory synaptic input.” Nat. Rev. Neurosci., 1, 181–190.

• Poirazi, P. & Mel, B. W. (2001). “Impact of active dendrites and structural plasticity on the memory capacity of neural tissue.” Neuron, 29, 779–796.

• Bliss, T. V. P. & Collingridge, G. L. (1993). “A synaptic model of memory: long-term potentiation in the hippocampus.” Nature, 361, 31–39.

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